1. bookVolume 6 (2021): Issue 1 (January 2021)
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On Some Integral Inequalities via Conformable Fractional Integrals

Published Online: 31 Dec 2020
Page range: 489 - 498
Received: 27 Nov 2019
Accepted: 23 Mar 2020
Journal Details
License
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Abstract

In the present note, we have given a new integral identity via Conformable fractional integrals and some further properties. We have proved some integral inequalities for different kinds of convexity via Conformable fractional integrals. We have also showed that special cases of our findings gave some new inequalities involving Riemann-Liouville fractional integrals.

Keywords

MSC 2010

Introduction

We will recall some preliminaries concepts to refresh our memories:

Definition 1

[See [5]] A mapping f : I → [0, ∞) is said to be logconvex or multiplicatively convex if log f is convex or equivalently for all θ, υI and ς ∈ [0, 1], one has the inequality: f(ςθ+(1ς)ϑ)[f(θ)]ς[f(ϑ)]1ς f\left( {\varsigma \theta + (1 - \varsigma )\vartheta } \right) \le {\left[ {f(\theta )} \right]^\varsigma }{\left[ {f(\vartheta )} \right]^{1 - \varsigma }}

We note that a log−convex function satisfy the condition of convexity, but the converse may not necessarily be true.

Definition 2

[See [6]] Let s ∈ (0, 1]. A function f : [0, ∞) → [0, ∞) is said to be sconvex in the second sense if f(tx+(1t)y)tsf(x)+(1t)sf(y) f\left( {tx + \left( {1 - t} \right)y} \right) \le {t^s}f\left( x \right) + {\left( {1 - t} \right)^s}f\left( y \right)

for all x, y ∈ ℝ+ and t ∈ [0, 1].

In [7], s−convexity introduced by Breckner as a generalization of convex functions. Also, Breckner proved the fact that the set valued map is s−convex only if the associated support function is s−convex function in [8]. Several properties of s−convexity in the first sense are discussed in the paper [6]. Obviously, s−convexity means just convexity when s = 1.

Definition 3

(See [5]) A function f : [a,b] → ℝ is said quasi-convex on [a,b] if f(λx+(1λ)y)max{f(x),f(y)},(QC) f\left( {\lambda x + (1 - \lambda )y} \right) \le \max \left\{ {f(x),f(y)} \right\},\;\;\;\;\left( {QC} \right) holds for all x,y ∈ [a,b] and λ ∈ [0, 1].

Clearly, any convex function is quasi-convex function. Furthermore, there exist quasi-convex functions which are not convex.

Let us recall the definition of Riemann-Liouville fractional integrals:

Definition 4

Let fL1[a,b]. The Riemann-Liouville integrals Ja+αf J_{a + }^\alpha f and Jbαf J_{b - }^\alpha f of order α > 0 are defined by Ja+αf(t)=1Γ(α)at(tx)α1f(x)dx,t>a J_{a + }^\alpha f(t) = {1 \over {\Gamma (\alpha )}}\int_a^t {(t - x)^{\alpha - 1}}f(x)dx,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} t > a and Jbαf(t)=1Γ(α)tb(xt)α1f(x)dx,t<b J_{b - }^\alpha f(t) = {1 \over {\Gamma (\alpha )}}\int_t^b {(x - t)^{\alpha - 1}}f(x)dx,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} t < b respectively where Γ(α)=0ettα1dt \Gamma (\alpha ) = \int_0^\infty {e^{ - t}}{t^{\alpha - 1}}dt . Here Ja+0f(t)=Jb0f(t)=f(t) J_{a + }^0f(t) = J_{b - }^0f(t) = f(t)

In the case of α = 1, the fractional integral reduces to classical integral.

We will mention the Beta function (See [4]): B(a,b)=Γ(a)Γ(b)Γ(a+b)=01ta1(1t)b1dt,a,b>0, B\left( {a,b} \right) = {{\Gamma (a)\Gamma (b)} \over {\Gamma (a + b)}} = \int_0^1 {t^{a - 1}}{\left( {1 - t} \right)^{b - 1}}dt,\;\;\;\;a,b > 0, where Γ(α)=0ettα1dt \Gamma \left( \alpha \right) = \int_0^\infty {e^{ - t}}{t^{\alpha - 1}}dt is Gamma function.

Incomplete Beta function is defined as: Bx(a,b)=0xta1(1t)b1dt,a,b>0. {B_x}\left( {a,b} \right) = \int_0^x {t^{a - 1}}{\left( {1 - t} \right)^{b - 1}}dt,\;\;\;\;a,b > 0.

In spite of its valuable contributions to mathematical analysis, the Riemann-Liouvile Fractional integrals have deficiencies. For example the solution of the differential equation that is given as; y(12)+y=x(12)+2Γ(2.5)x(32),y(0)=0 {y^{({1 \over 2})}} + y = {x^{({1 \over 2})}} + {2 \over {\Gamma (2.5)}}{x^{({3 \over 2})}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} y(0) = 0 where y(12) {y^{({1 \over 2})}} is the fractional derivative of y of order 12 {1 \over 2} .

The solution of the above differential equation have caused to imagine on a new and simple represention of the definition of fractional derivative. In [2], Khalil et al. gave a new definition that is called “conformable fractional derivative”. They not only proved further properties of this definitons but also gave the differences with the other fractional derivatives. Besides, another considerable study have presented by Abdeljawad to discuss the basic concepts of fractional calculus. In [1], Abdeljawad gave the following definitions of Right-Left conformable fractional integrals:

Definition 5

Let α ∈ (n, n + 1], n = 0, 1, 2,... and set β = αn. Then the left conformable fractional integral of any order α > 0 is defined by (Iαaf)(t)=1n!at(tx)n(xa)β1f(x)dx (I_\alpha ^af)(t) = {1 \over {n!}}\int_a^t {(t - x)^n}{(x - a)^{\beta - 1}}f(x)dx

Definition 6

Analogously, the right conformable fractional integral of any order α > 0 is defined by (bIαf)(t)=1n!tb(xt)n(bx)β1f(x)dx. {(^b}{I_\alpha }f)(t) = {1 \over {n!}}\int_t^b {(x - t)^n}{(b - x)^{\beta - 1}}f(x)dx.

Notice that if α = n + 1 then β = αn = n + 1 − n = 1, hence (In+1af)(t)=(Ja+n+1f)(t) (I_{n + 1}^af)(t) = (J_{a + }^{n + 1}f)(t) and (bIn+1f)(t)=(Jbn+1f)(t) {(^b}{I_{n + 1}}f)(t) = \left( {J_{b - }^{n + 1}f} \right)(t) .

In [1] and [2], authors have pointed that the Riemann-Liouville derivatives are not valid for product of two functions. In this case, the inequalities that have been proved by Riemann-Liouville integrals are not valid. The results which are obtained by using the conformable fractional integrals have a wide range of validity. (Let us consider the function f defined as f : ℝ+ ℝ, f = x2ex which is convex.). The interested readers can find several new integral inequalities via different fractional integral operators in the papers [9,10,11,12,13,14].

In this paper, some new integral inequalities have been proved by using conformable fractional integrals for functions whose derivatives of absolute values are quasi-convex, s−convex and log−convex functions.

Main Results

In order to prove our main theorems, we need the following lemma.

Lemma 1

(See [3]) Let f : [a,b] ⊂ ℝ → ℝ be a differentiable mapping on (a, b) such that fL1 [a, b]. Then for all x ∈ [a, b] and α ∈ (n, n + 1], the following equality holds: (xa)α+1n!(ba)[01Bt(n+1,αn)2f'(1+t2x+1t2a)dt01Bt(n+1,αn)2f'(1t2x+1+t2a)dt](bx)α+1n!(ba)[01Bt(n+1,αn)2f'(1+t2x+1t2b)dt+01Bt(n+1,αn)2f'(1t2x+1+t2b)dt]=Γ(αn)(xa)αΓ(α+1)[f(x)+f(a)](ba)2αba[xIαf(x+a2)+Iαaf(x+a2)]+Γ(αn)(bx)αΓ(α+1)[f(x)+f(b)](ba)2αba[xIαf(x+b2)+Iαbf(x+b2)] \matrix{ {} \hfill & {{{{{(x - a)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\left[ {\int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 + t} \over 2}x + {{1 - t} \over 2}a} \right)dt} \right.} \hfill \cr {} \hfill & {\left. { - \int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 - t} \over 2}x + {{1 + t} \over 2}a} \right)dt} \right]} \hfill \cr {} \hfill & { - {{{{(b - x)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\left[ {\int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 + t} \over 2}x + {{1 - t} \over 2}b} \right)dt} \right.} \hfill \cr {} \hfill & {\left. { + \int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 - t} \over 2}x + {{1 + t} \over 2}b} \right)dt} \right]} \hfill \cr = \hfill & {{{\Gamma \left( {\alpha - n} \right){{\left( {x - a} \right)}^\alpha }} \over {\Gamma \left( {\alpha + 1} \right)}}{{\left[ {f\left( x \right) + f\left( a \right)} \right]} \over {\left( {b - a} \right)}} - {{{2^\alpha }} \over {b - a}}\left[ {^x{I_\alpha }f\left( {{{x + a} \over 2}} \right) + I_\alpha ^af\left( {{{x + a} \over 2}} \right)} \right]} \hfill \cr {} \hfill & { + {{\Gamma \left( {\alpha - n} \right){{\left( {b - x} \right)}^\alpha }} \over {\Gamma \left( {\alpha + 1} \right)}}{{\left[ {f\left( x \right) + f\left( b \right)} \right]} \over {\left( {b - a} \right)}} - {{{2^\alpha }} \over {b - a}}\left[ {^x{I_\alpha }f\left( {{{x + b} \over 2}} \right) + I_\alpha ^bf\left( {{{x + b} \over 2}} \right)} \right]} \hfill \cr } where Bt (a,b) is incompleted beta function.

Proof

By using the integration by parts formula in the left hand side of the above equality, the right hand side can be obtained. The details of the proof are left to the interested reader.

For the simplicity, in the sequel of the paper, we will use following notation Ff(α,n;x)=Γ(αn)(xa)αΓ(α+1)[f(x)+f(a)](ba)2αba[xIαf(x+a2)+Iαaf(x+a2)]+Γ(αn)(bx)αΓ(α+1)[f(x)+f(b)](ba)2αba[xIαf(x+b2)+Iαbf(x+b2)] \matrix{ {} \hfill & {{F_f}(\alpha ,n;x)} \hfill \cr = \hfill & {{{\Gamma \left( {\alpha - n} \right){{\left( {x - a} \right)}^\alpha }} \over {\Gamma \left( {\alpha + 1} \right)}}{{\left[ {f\left( x \right) + f\left( a \right)} \right]} \over {\left( {b - a} \right)}} - {{{2^\alpha }} \over {b - a}}\left[ {^x{I_\alpha }f\left( {{{x + a} \over 2}} \right) + I_\alpha ^af\left( {{{x + a} \over 2}} \right)} \right]} \hfill \cr {} \hfill & { + {{\Gamma \left( {\alpha - n} \right){{\left( {b - x} \right)}^\alpha }} \over {\Gamma \left( {\alpha + 1} \right)}}{{\left[ {f\left( x \right) + f\left( b \right)} \right]} \over {\left( {b - a} \right)}} - {{{2^\alpha }} \over {b - a}}\left[ {^x{I_\alpha }f\left( {{{x + b} \over 2}} \right) + I_\alpha ^bf\left( {{{x + b} \over 2}} \right)} \right]} \hfill \cr }

The next theorems give new results of conformable fractional integrals by means of Lemma 1 for quasi-convex, s−convex, m−convex and log−convex functions.

Theorem 1

Let f : [a, b] ⊂ ℝ → ℝ be a differentiable mapping on (a, b) such that fL1 [a, b]. If | f | is quasi-convex on [a, b], then the following inequality holds for conformable fractional integrals: |Ff(α,n;x)|Γ(αn+1)Γ(α+2)(ba)((xa)α+1sup{|f'(x)|,|f'(a)|}+(bx)α+1sup{|f'(x)|,|f'(b)|}) \matrix{ {} \hfill & {\left| {{F_f}(\alpha ,n;x)} \right|} \hfill \cr \le \hfill & {{{\Gamma \left( {\alpha - n + 1} \right)} \over {\Gamma \left( {\alpha + 2} \right)\left( {b - a} \right)}}\left( {{{(x - a)}^{\alpha + 1}}\sup \left\{ {\left| {{f^\prime}\left( x \right)} \right|,\left| {{f^\prime}\left( a \right)} \right|} \right\} + {{(b - x)}^{\alpha + 1}}\sup \left\{ {\left| {{f^\prime}\left( x \right)} \right|,\left| {{f^\prime}\left( b \right)} \right|} \right\}} \right)} \hfill \cr } with α ∈ (n,n + 1], n = 0, 1, 2...

Proof

Since | f | is quasi-convex on [a, b] and by Lemma 1, we can write I1=|(xa)α+1n!(ba)[01Bt(n+1,αn)2f'(1+t2x+1t2a)dt01Bt(n+1,αn)2f'(1t2x+1+t2a)dt]|(xa)α+1n!(ba)[01|Bt(n+1,αn)2||f'(1+t2x+1t2a)|dt+01|Bt(n+1,αn)2||f'(1t2x+1+t2a)|dt](xa)α+1n!(ba)sup{|f'(x)|,|f'(a)|}01|Bt(n+1,αn)|dt \matrix{ {{I_1}} \hfill & = \hfill & {\left| {{{{{(x - a)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\left[ {\int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 + t} \over 2}x + {{1 - t} \over 2}a} \right)dt} \right.} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. {\left. { - \int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 - t} \over 2}x + {{1 + t} \over 2}a} \right)dt} \right]} \right|} \hfill \cr {} \hfill & {} \hfill \le & { {{{{(x - a)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\left[ {\int_0^1 \left| {{{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}} \right|\left| {{f^\prime}\left( {{{1 + t} \over 2}x + {{1 - t} \over 2}a} \right)} \right|dt} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. { + \int_0^1 \left| {{{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}} \right|\left| {{f^\prime}\left( {{{1 - t} \over 2}x + {{1 + t} \over 2}a} \right)} \right|dt} \right]} \hfill \cr {} \hfill & {} \hfill \le & { {{{{(x - a)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\sup \left\{ {\left| {{f^\prime}\left( x \right)} \right|,\left| {{f^\prime}\left( a \right)} \right|} \right\}\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|dt} \hfill \cr } Similarly, we obtain I2=|(bx)α+1n!(ba)[01Bt(n+1,αn)2f'(1+t2x+1t2b)dt01Bt(n+1,αn)2f'(1t2x+1+t2b)dt]|(bx)α+1n!(ba)sup{|f'(x)|,|f'(b)|}01|Bt(n+1,αn)|dt \matrix{ {{I_2}} \hfill & = \hfill & {\left| {{{{{(b - x)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\left[ {\int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 + t} \over 2}x + {{1 - t} \over 2}b} \right)dt} \right.} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. {\left. { - \int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 - t} \over 2}x + {{1 + t} \over 2}b} \right)dt} \right]} \right|} \hfill \cr {} \hfill & {} \hfill \le & { {{{{(b - x)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\sup \left\{ {\left| {{f^\prime}\left( x \right)} \right|,\left| {{f^\prime}\left( b \right)} \right|} \right\}\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|dt} \hfill \cr }

We apply the integration by parts formula to incomplated beta function 01|Bt(n+1,αn)|dt=B(n+1,αn)B(n+2,αn)=n!Γ(αn+1)Γ(α+2) \matrix{ {\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|dt} \hfill & = \hfill & {B\left( {n + 1,\alpha - n} \right) - B\left( {n + 2,\alpha - n} \right)} \hfill \cr {} \hfill & = \hfill & {{{n!\Gamma \left( {\alpha - n + 1} \right)} \over {\Gamma \left( {\alpha + 2} \right)}}} \hfill \cr }

Adding the above quantities lead to Theorem 1 and the proof is completed.

Remark 1

Under conditions of Theorem 1, if α = n + 1, then |(xa)n+1Γ(n+2)(ba)[f'(x)+f'(a)]2n+1(ba)[Jxn+1+Ja+n+1]+(bx)n+1Γ(n+2)(ba)[f'(x)+f'(b)]2n+1(ba)[Jxn+1+Jb+n+1]|(xa)n+2sup{|f'(x)|,|f'(a)|}+(bx)n+2sup{|f'(x)|,|f'(b)|}(n+2)!(ba) \matrix{ {} \hfill & {\left| {{{{{\left( {x - a} \right)}^{n + 1}}} \over {\Gamma \left( {n + 2} \right)\left( {b - a} \right)}}\left[ {{f^\prime}\left( x \right) + {f^\prime}\left( a \right)} \right] - {{{2^{n + 1}}} \over {\left( {b - a} \right)}}\left[ {J_{x - }^{n + 1} + J_{a + }^{n + 1}} \right]} \right.} \hfill \cr {} \hfill & {\left. { + {{{{\left( {b - x} \right)}^{n + 1}}} \over {\Gamma \left( {n + 2} \right)\left( {b - a} \right)}}\left[ {{f^\prime}\left( x \right) + {f^\prime}\left( b \right)} \right] - {{{2^{n + 1}}} \over {\left( {b - a} \right)}}\left[ {J_{x - }^{n + 1} + J_{b + }^{n + 1}} \right]} \right|} \hfill \cr \le \hfill & {{{{{(x - a)}^{n + 2}}\sup \left\{ {\left| {{f^\prime}\left( x \right)} \right|,\left| {{f^\prime}\left( a \right)} \right|} \right\} + {{(b - x)}^{n + 2}}\sup \left\{ {\left| {{f^\prime}\left( x \right)} \right|,\left| {{f^\prime}\left( b \right)} \right|} \right\}} \over {\left( {n + 2} \right)!\left( {b - a} \right)}}} \hfill \cr }

Theorem 2

Let f : [a, b] ⊂ ℝ → ℝ be a differentiable mapping on (a, b) such that fL1 [a, b]. If | f | is sconvex in the second sense with s ∈ (0, 1], then the following inequality holds for conformable fractional integrals: |Ff(α,n;x)|(A1+A2)2s+1n!(ba)[(xa)α+1(|f'(x)|+|f'(a)|)+(bx)α+1(|f'(x)|+|f'(b)|)] \matrix{ {} \hfill & {\left| {{F_f}(\alpha ,n;x)} \right|} \hfill \cr \le \hfill & {{{\left( {{{\cal A}_1} + {{\cal A}_2}} \right)} \over {{2^{s + 1}}n!\left( {b - a} \right)}}\left[ {{{(x - a)}^{\alpha + 1}}\left( {\left| {{f^\prime}\left( x \right)} \right| + \left| {{f^\prime}\left( a \right)} \right|} \right) + {{(b - x)}^{\alpha + 1}}\left( {\left| {{f^\prime}\left( x \right)} \right| + \left| {{f^\prime}\left( b \right)} \right|} \right)} \right]} \hfill \cr } with α ∈ (n,n + 1], n = 0, 1, 2... where 𝒜1 and 𝒜2 are given as: A1=01|Bt(n+1,αn)|(1+t)sdt=Γ(n+1)Γ(αn)×[2sΓ(α+1)2F1(n+1,1s;α+1;1)Γ(α+1)(s+1)] \matrix{ {{{\cal A}_1}} \hfill & = \hfill & {\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left( {1 + t} \right)}^s}dt = \Gamma \left( {n + 1} \right)\Gamma \left( {\alpha - n} \right)} \hfill \cr {} \hfill & {} \hfill & { \times \left[ {{{{2^s} - \Gamma {{\left( {\alpha + 1} \right)}_2}{F_1}\left( {n + 1, - 1 - s;\alpha + 1; - 1} \right)} \over {\Gamma \left( {\alpha + 1} \right)\left( {s + 1} \right)}}} \right]} \hfill \cr } and A2=01|Bt(n+1,αn)|(1t)sdt=B(n+1,αn+s+1)(s+1). \matrix{ {{{\cal A}_2}} \hfill & = \hfill & {\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left( {1 - t} \right)}^s}dt} \hfill \cr {} \hfill & = \hfill & {{{B\left( {n + 1,\alpha - n + s + 1} \right)} \over {\left( {s + 1} \right)}}.} \hfill \cr }

Proof

Since |f | is s−convex on [a, b] and by Lemma 1, we can write Q1=|(xa)α+1n!(ba)[01Bt(n+1,αn)2f'(1+t2x+1t2a)dt01Bt(n+1,αn)2f'(1t2x+1+t2a)dt]|(xa)α+1n!(ba)[01|Bt(n+1,αn)2||f'(1+t2x+1t2a)|dt+01|Bt(n+1,αn)2||f'(1t2x+1+t2a)|dt](xa)α+12n!(ba)[01|Bt(n+1,αn)|{(1+t2)s|f'(x)|+(1t2)s|f'(a)|}dt+01|Bt(n+1,αn)|{(1t2)s|f'(x)|+(1+t2)s|f'(a)|}dt]=(xa)α+12s+1n!(ba)[|f'(x)|01|Bt(n+1,αn)|(1+t)sdt+|f'(a)|01|Bt(n+1,αn)|(1t)sdt+|f'(x)|01|Bt(n+1,αn)|(1t)sdt+|f'(a)|01|Bt(n+1,αn)|(1+t)sdt] \matrix{ {{Q_1}} \hfill & = \hfill & {\left| {{{{{(x - a)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\left[ {\int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 + t} \over 2}x + {{1 - t} \over 2}a} \right)dt} \right.} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. {\left. { - \int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 - t} \over 2}x + {{1 + t} \over 2}a} \right)dt} \right]} \right|} \hfill \cr {} \hfill & \le \hfill & {{{{{(x - a)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\left[ {\int_0^1 \left| {{{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}} \right|\left| {{f^\prime}\left( {{{1 + t} \over 2}x + {{1 - t} \over 2}a} \right)} \right|dt} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. { + \int_0^1 \left| {{{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}} \right|\left| {{f^\prime}\left( {{{1 - t} \over 2}x + {{1 + t} \over 2}a} \right)} \right|dt} \right]} \hfill \cr {} \hfill & \le \hfill & {{{{{(x - a)}^{\alpha + 1}}} \over {2n!\left( {b - a} \right)}}\left[ {\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|\left\{ {{{\left( {{{1 + t} \over 2}} \right)}^s}\left| {{f^\prime}\left( x \right)} \right| + {{\left( {{{1 - t} \over 2}} \right)}^s}\left| {{f^\prime}\left( a \right)} \right|} \right\}dt} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. { + \int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|\left\{ {{{\left( {{{1 - t} \over 2}} \right)}^s}\left| {{f^\prime}\left( x \right)} \right| + {{\left( {{{1 + t} \over 2}} \right)}^s}\left| {{f^\prime}\left( a \right)} \right|} \right\}dt} \right]} \hfill \cr {} \hfill & = \hfill & {{{{{(x - a)}^{\alpha + 1}}} \over {{2^{s + 1}}n!\left( {b - a} \right)}}\left[ {\left| {{f^\prime}\left( x \right)} \right|\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left( {1 + t} \right)}^s}dt + \left| {{f^\prime}\left( a \right)} \right|\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left( {1 - t} \right)}^s}dt} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. { + \left| {{f^\prime}\left( x \right)} \right|\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left( {1 - t} \right)}^s}dt + \left| {{f^\prime}\left( a \right)} \right|\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left( {1 + t} \right)}^s}dt} \right]} \hfill \cr }

Similarly, we have Q2=|(bx)α+1n!(ba)[01Bt(n+1,αn)2f'(1+t2x+1t2b)dt01Bt(n+1,αn)2f'(1t2x+1+t2b)dt]|(bx)α+12s+1n!(ba)[|f'(x)|01|Bt(n+1,αn)|(1+t)sdt+|f'(b)|01|Bt(n+1,αn)|(1t)sdt+|f'(x)|01|Bt(n+1,αn)|(1t)sdt+|f'(b)|01|Bt(n+1,αn)|(1+t)sdt] \matrix{ {{Q_2}} \hfill & = \hfill & {\left| {{{{{(b - x)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\left[ {\int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 + t} \over 2}x + {{1 - t} \over 2}b} \right)dt} \right.} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. {\left. { - \int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 - t} \over 2}x + {{1 + t} \over 2}b} \right)dt} \right]} \right|} \hfill \cr {} \hfill & \le \hfill & {{{{{(b - x)}^{\alpha + 1}}} \over {{2^{s + 1}}n!\left( {b - a} \right)}}\left[ {\left| {{f^\prime}\left( x \right)} \right|\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left( {1 + t} \right)}^s}dt} \right.} \hfill \cr {} \hfill & {} \hfill & { + \left| {{f^\prime}\left( b \right)} \right|\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left( {1 - t} \right)}^s}dt} \hfill \cr {} \hfill & {} \hfill & {\left. { + \left| {{f^\prime}\left( x \right)} \right|\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left( {1 - t} \right)}^s}dt + \left| {{f^\prime}\left( b \right)} \right|\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left( {1 + t} \right)}^s}dt} \right]} \hfill \cr }

Substituting 𝒜1 and 𝒜2 into Q1 and Q2 inequalities and simplifying lead to the required inequality. The proof of Theorem 2 is completed.

Remark 2

Under conditions of Theorem 2,

if s = 1, then |Ff(α,n;x)|14n!(ba)[2B(n+1,αn)B(n+1,αn1)B(n+2,αn)12B(n+3,αn)]×[(xa)α+1(|f'(x)|+|f'(a)|)+(bx)α+1(|f'(x)|+|f'(b)|)] \matrix{ {\left| {{F_f}(\alpha ,n;x)} \right| \le {1 \over {4n!\left( {b - a} \right)}}\left[ {2B\left( {n + 1,\alpha - n} \right) - B\left( {n + 1,\alpha - n - 1} \right) - B\left( {n + 2,\alpha - n} \right) - {1 \over 2}B\left( {n + 3,\alpha - n} \right)} \right]} \hfill \cr { \times \left[ {{{(x - a)}^{\alpha + 1}}\left( {\left| {{f^\prime}\left( x \right)} \right| + \left| {{f^\prime}\left( a \right)} \right|} \right) + {{(b - x)}^{\alpha + 1}}\left( {\left| {{f^\prime}\left( x \right)} \right| + \left| {{f^\prime}\left( b \right)} \right|} \right)} \right]} \hfill \cr }

if α = n + 1, then |(xa)n+1Γ(n+2)(ba)[f'(x)+f'(a)]2n+1(ba)[Jxn+1+Ja+n+1]+(bx)n+1Γ(n+2)(ba)[f'(x)+f'(b)]2n+1(ba)[Jxn+1+Jb+n+1]|12s+1(s+1)(ba)[(2sΓ(n+2)2F1(n+1,1s;n+2;1)Γ(n+2)+Γ(s+2)Γ(n+s+3))]×[(xa)n+2(|f'(x)|+|f'(a)|)+(bx)n+2(|f'(x)|+|f'(b)|)] \matrix{ {} \hfill & {\left| {{{{{\left( {x - a} \right)}^{n + 1}}} \over {\Gamma \left( {n + 2} \right)\left( {b - a} \right)}}\left[ {{f^\prime}\left( x \right) + {f^\prime}\left( a \right)} \right] - {{{2^{n + 1}}} \over {\left( {b - a} \right)}}\left[ {J_{x - }^{n + 1} + J_{a + }^{n + 1}} \right]} \right.} \hfill \cr {} \hfill & {\left. { + {{{{\left( {b - x} \right)}^{n + 1}}} \over {\Gamma \left( {n + 2} \right)\left( {b - a} \right)}}\left[ {{f^\prime}\left( x \right) + {f^\prime}\left( b \right)} \right] - {{{2^{n + 1}}} \over {\left( {b - a} \right)}}\left[ {J_{x - }^{n + 1} + J_{b + }^{n + 1}} \right]} \right|} \hfill \cr \le \hfill & {{1 \over {{2^{s + 1}}\left( {s + 1} \right)\left( {b - a} \right)}}\left[ {\left( {{{{2^s} - \Gamma {{\left( {n + 2} \right)}_2}{F_1}\left( {n + 1, - 1 - s;n + 2; - 1} \right)} \over {\Gamma \left( {n + 2} \right)}} + {{\Gamma \left( {s + 2} \right)} \over {\Gamma \left( {n + s + 3} \right)}}} \right)} \right]} \hfill \cr {} \hfill & { \times \left[ {{{\left( {x - a} \right)}^{n + 2}}\left( {\left| {{f^\prime}\left( x \right)} \right| + \left| {{f^\prime}\left( a \right)} \right|} \right) + {{\left( {b - x} \right)}^{n + 2}}\left( {\left| {{f^\prime}\left( x \right)} \right| + \left| {{f^\prime}\left( b \right)} \right|} \right)} \right]} \hfill \cr }

if α = n + 1, s = 1, we have |(xa)n+1Γ(n+2)(ba)[f'(x)+f'(a)]2n+1(ba)[Jxn+1+Ja+n+1]+(bx)n+1Γ(n+2)(ba)[f'(x)+f'(b)]2n+1(ba)[Jxn+1+Jb+n+1]|(xa)n+24n!(ba)(|f'(x)|+|f'(a)|)×(3(n+2)(n+3)2(n+1)(n+3)(n+1)(n+2)+n(n+3)2(n+1)(n+2)(n+3))+(bx)n+24n!(ba)(|f'(x)|+|f'(b)|)(3(n+2)(n+3)2(n+1)(n+3)(n+1)(n+2)+n(n+3)2(n+1)(n+2)(n+3)) \matrix{ {} \hfill & {\left| {{{{{\left( {x - a} \right)}^{n + 1}}} \over {\Gamma \left( {n + 2} \right)\left( {b - a} \right)}}\left[ {{f^\prime}\left( x \right) + {f^\prime}\left( a \right)} \right] - {{{2^{n + 1}}} \over {\left( {b - a} \right)}}\left[ {J_{x - }^{n + 1} + J_{a + }^{n + 1}} \right]} \right.} \hfill \cr {} \hfill & {\left. { + {{{{\left( {b - x} \right)}^{n + 1}}} \over {\Gamma \left( {n + 2} \right)\left( {b - a} \right)}}\left[ {{f^\prime}\left( x \right) + {f^\prime}\left( b \right)} \right] - {{{2^{n + 1}}} \over {\left( {b - a} \right)}}\left[ {J_{x - }^{n + 1} + J_{b + }^{n + 1}} \right]} \right|} \hfill \cr \le \hfill & {{{{{(x - a)}^{n + 2}}} \over {4n!\left( {b - a} \right)}}\left( {\left| {{f^\prime}\left( x \right)} \right| + \left| {{f^\prime}\left( a \right)} \right|} \right)} \hfill \cr {} \hfill & { \times \left( {{{3\left( {n + 2} \right)\left( {n + 3} \right) - 2\left( {n + 1} \right)\left( {n + 3} \right) - \left( {n + 1} \right)\left( {n + 2} \right) + n\left( {n + 3} \right)} \over {2\left( {n + 1} \right)\left( {n + 2} \right)\left( {n + 3} \right)}}} \right)} \hfill \cr {} \hfill & { + {{{{(b - x)}^{n + 2}}} \over {4n!\left( {b - a} \right)}}\left( {\left| {{f^\prime}\left( x \right)} \right| + \left| {{f^\prime}\left( b \right)} \right|} \right)\left( {{{3\left( {n + 2} \right)\left( {n + 3} \right) - 2\left( {n + 1} \right)\left( {n + 3} \right) - \left( {n + 1} \right)\left( {n + 2} \right) + n\left( {n + 3} \right)} \over {2\left( {n + 1} \right)\left( {n + 2} \right)\left( {n + 3} \right)}}} \right)} \hfill \cr }

Theorem 3

Let f : [a, b] ⊂ ℝ → ℝ be a differentiable mapping on (a, b) such that f L1 [a, b]. If | f | is log−convex on [a, b], then the following inequality holds for conformable fractional integrals: |Ff(α,n;x)|B(n+1,αn)|f'(x)|122n!(ba)[(xa)α+1|f'(a)|12(Ω1+Ω2)+(bx)α+1|f'(b)|12(Ω3+Ω4)] \left| {{F_f}(\alpha ,n;x)} \right| \le {{B\left( {n + 1,\alpha - n} \right){{\left| {{f^\prime}\left( x \right)} \right|}^{{1 \over 2}}}} \over {2n!\left( {b - a} \right)}}\left[ {{{\left( {x - a} \right)}^{\alpha + 1}}{{\left| {{f^\prime}\left( a \right)} \right|}^{{1 \over 2}}}\left( {{\Omega _1} + {\Omega _2}} \right) + {{\left( {b - x} \right)}^{\alpha + 1}}{{\left| {{f^\prime}\left( b \right)} \right|}^{{1 \over 2}}}\left( {{\Omega _3} + {\Omega _4}} \right)} \right] where B(a, b) Euler beta function with α ∈ (n, n + 1], n = 0, 1, 2...

Proof

Since | f | is log−convex on [a, b] and by Lemma 1, we can write Ψ1=|(xa)α+1n!(ba)[01Bt(n+1,αn)2f'(1+t2x+1t2a)dt01Bt(n+1,αn)2f'(1t2x+1+t2a)dt]|(xa)α+1n!(ba)[01|Bt(n+1,αn)2||f'(1+t2x+1t2a)|dt+01|Bt(n+1,αn)2||f'(1t2x+1+t2a)|dt](xa)α+12n!(ba)[01|Bt(n+1,αn)||f'(x)|(1+t2)|f'(a)|(1t2)dt+01|Bt(n+1,αn)||f'(x)|(1t2)|f'(a)|(1+t2)dt]=(xa)α+12n!(ba)|f'(x)|12|f'(a)|12[01|Bt(n+1,αn)|[|f'(x)||f'(a)|]t2dt+01|Bt(n+1,αn)|[|f'(a)||f'(x)|]t2dt] \matrix{ {{\Psi _1}} \hfill & = \hfill & {\left| {{{{{(x - a)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\left[ {\int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 + t} \over 2}x + {{1 - t} \over 2}a} \right)dt} \right.} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. {\left. { - \int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 - t} \over 2}x + {{1 + t} \over 2}a} \right)dt} \right]} \right|} \hfill \cr {} \hfill & \le \hfill & {{{{{(x - a)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\left[ {\int_0^1 \left| {{{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}} \right|\left| {{f^\prime}\left( {{{1 + t} \over 2}x + {{1 - t} \over 2}a} \right)} \right|dt} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. { + \int_0^1 \left| {{{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}} \right|\left| {{f^\prime}\left( {{{1 - t} \over 2}x + {{1 + t} \over 2}a} \right)} \right|dt} \right]} \hfill \cr {} \hfill & \le \hfill & {{{{{(x - a)}^{\alpha + 1}}} \over {2n!\left( {b - a} \right)}}\left[ {\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left| {{f^\prime}\left( x \right)} \right|}^{\left( {{{1 + t} \over 2}} \right)}}{{\left| {{f^\prime}\left( a \right)} \right|}^{\left( {{{1 - t} \over 2}} \right)}}dt} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. { + \int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left| {{f^\prime}\left( x \right)} \right|}^{\left( {{{1 - t} \over 2}} \right)}}{{\left| {{f^\prime}\left( a \right)} \right|}^{\left( {{{1 + t} \over 2}} \right)}}dt} \right]} \hfill \cr {} \hfill & = \hfill & {{{{{(x - a)}^{\alpha + 1}}} \over {2n!\left( {b - a} \right)}}{{\left| {{f^\prime}\left( x \right)} \right|}^{{1 \over 2}}}{{\left| {{f^\prime}\left( a \right)} \right|}^{{1 \over 2}}}\left[ {\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left[ {{{\left| {{f^\prime}\left( x \right)} \right|} \over {\left| {{f^\prime}\left( a \right)} \right|}}} \right]}^{{t \over 2}}}dt} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. { + \int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left[ {{{\left| {{f^\prime}\left( a \right)} \right|} \over {\left| {{f^\prime}\left( x \right)} \right|}}} \right]}^{{t \over 2}}}dt} \right]} \hfill \cr }

Similarly, we have Ψ2=|(bx)α+1n!(ba)[01Bt(n+1,αn)2f'(1+t2x+1t2b)dt01Bt(n+1,αn)2f'(1t2x+1+t2b)dt]|(bx)α+1n!(ba)[01|Bt(n+1,αn)2||f'(1+t2x+1t2b)|dt+01|Bt(n+1,αn)2||f'(1t2x+1+t2b)|dt](bx)α+12n!(ba)|f'(x)|12|f'(b)|12[01|Bt(n+1,αn)|[|f'(x)||f'(b)|]t2dt+01|Bt(n+1,αn)|[|f'(b)||f'(x)|]t2dt] \matrix{ {{\Psi _2}} \hfill & = \hfill & {\left| {{{{{(b - x)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\left[ {\int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 + t} \over 2}x + {{1 - t} \over 2}b} \right)dt} \right.} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. {\left. { - \int_0^1 {{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}{f^\prime}\left( {{{1 - t} \over 2}x + {{1 + t} \over 2}b} \right)dt} \right]} \right|} \hfill \cr {} \hfill & \le \hfill & {{{{{(b - x)}^{\alpha + 1}}} \over {n!\left( {b - a} \right)}}\left[ {\int_0^1 \left| {{{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}} \right|\left| {{f^\prime}\left( {{{1 + t} \over 2}x + {{1 - t} \over 2}b} \right)} \right|dt} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. { + \int_0^1 \left| {{{{B_t}\left( {n + 1,\alpha - n} \right)} \over 2}} \right|\left| {{f^\prime}\left( {{{1 - t} \over 2}x + {{1 + t} \over 2}b} \right)} \right|dt} \right]} \hfill \cr {} \hfill & \le \hfill & {{{{{(b - x)}^{\alpha + 1}}} \over {2n!\left( {b - a} \right)}}{{\left| {{f^\prime}\left( x \right)} \right|}^{{1 \over 2}}}{{\left| {{f^\prime}\left( b \right)} \right|}^{{1 \over 2}}}\left[ {\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left[ {{{\left| {{f^\prime}\left( x \right)} \right|} \over {\left| {{f^\prime}\left( b \right)} \right|}}} \right]}^{{t \over 2}}}dt} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. { + \int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left[ {{{\left| {{f^\prime}\left( b \right)} \right|} \over {\left| {{f^\prime}\left( x \right)} \right|}}} \right]}^{{t \over 2}}}dt} \right]} \hfill \cr }

Substituting Ω1=01|Bt(n+1,αn)|[|f'(x)||f'(a)|]t2dt=|f'(x)||f'(a)|ln|f'(x)||f'(a)|B(n+1,αn)1ln|f'(x)||f'(a)|01tn(1t)αn1(|f'(x)||f'(a)|)tdtΩ2=01|Bt(n+1,αn)|[|f'(a)||f'(x)|]t2dt=|f'(a)||f'(x)|ln|f'(a)||f'(x)|B(n+1,αn)1ln|f'(a)||f'(x)|01tn(1t)αn1(|f'(a)||f'(x)|)tdtΩ3=01|Bt(n+1,αn)|[|f'(x)||f'(b)|]t2dt=|f'(x)||f'(b)|ln|f'(x)||f'(b)|B(n+1,αn)1ln|f'(x)||f'(b)|01tn(1t)αn1(|f'(x)||f'(b)|)tdtΩ4=01|Bt(n+1,αn)|[|f'(b)||f'(x)|]t2dt=|f'(b)||f'(x)|ln|f'(b)||f'(x)|B(n+1,αn)1ln|f'(b)||f'(x)|01tn(1t)αn1(|f'(b)||f'(x)|)tdt \matrix{ {{\Omega _1}} \hfill & = \hfill & {\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left[ {{{\left| {{f^\prime}\left( x \right)} \right|} \over {\left| {{f^\prime}\left( a \right)} \right|}}} \right]}^{{t \over 2}}}dt} \hfill \cr {} \hfill & = \hfill & {{{\sqrt {{{\left| {{f^\prime}\left( x \right)} \right|} \over {\left| {{f^\prime}\left( a \right)} \right|}}} } \over {\ln {{\left| {{f^\prime}\left( x \right)} \right|} \over {\left| {{f^\prime}\left( a \right)} \right|}}}}B\left( {n + 1,\alpha - n} \right) - {1 \over {\ln \sqrt {{{\left| {{f^\prime}\left( x \right)} \right|} \over {\left| {{f^\prime}\left( a \right)} \right|}}} }}\int_0^1 {t^n}{{\left( {1 - t} \right)}^{\alpha - n - 1}}{{\left( {\sqrt {{{\left| {{f^\prime}\left( x \right)} \right|} \over {\left| {{f^\prime}\left( a \right)} \right|}}} } \right)}^t}dt} \hfill \cr {{\Omega _2}} \hfill & = \hfill & {\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left[ {{{\left| {{f^\prime}\left( a \right)} \right|} \over {\left| {{f^\prime}\left( x \right)} \right|}}} \right]}^{{t \over 2}}}dt} \hfill \cr {} \hfill & = \hfill & {{{\sqrt {{{\left| {{f^\prime}\left( a \right)} \right|} \over {\left| {{f^\prime}\left( x \right)} \right|}}} } \over {\ln {{\left| {{f^\prime}\left( a \right)} \right|} \over {\left| {{f^\prime}\left( x \right)} \right|}}}}B\left( {n + 1,\alpha - n} \right) - {1 \over {\ln \sqrt {{{\left| {{f^\prime}\left( a \right)} \right|} \over {\left| {{f^\prime}\left( x \right)} \right|}}} }}\int_0^1 {t^n}{{\left( {1 - t} \right)}^{\alpha - n - 1}}{{\left( {\sqrt {{{\left| {{f^\prime}\left( a \right)} \right|} \over {\left| {{f^\prime}\left( x \right)} \right|}}} } \right)}^t}dt} \hfill \cr {{\Omega _{_3}}} \hfill & = \hfill & {\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left[ {{{\left| {{f^\prime}\left( x \right)} \right|} \over {\left| {{f^\prime}\left( b \right)} \right|}}} \right]}^{{t \over 2}}}dt} \hfill \cr {} \hfill & = \hfill & {{{\sqrt {{{\left| {{f^\prime}\left( x \right)} \right|} \over {\left| {{f^\prime}\left( b \right)} \right|}}} } \over {\ln {{\left| {{f^\prime}\left( x \right)} \right|} \over {\left| {{f^\prime}\left( b \right)} \right|}}}}B\left( {n + 1,\alpha - n} \right) - {1 \over {\ln \sqrt {{{\left| {{f^\prime}\left( x \right)} \right|} \over {\left| {{f^\prime}\left( b \right)} \right|}}} }}\int_0^1 {t^n}{{\left( {1 - t} \right)}^{\alpha - n - 1}}{{\left( {\sqrt {{{\left| {{f^\prime}\left( x \right)} \right|} \over {\left| {{f^\prime}\left( b \right)} \right|}}} } \right)}^t}dt} \hfill \cr {{\Omega _4}} \hfill & = \hfill & {\int_0^1 \left| {{B_t}\left( {n + 1,\alpha - n} \right)} \right|{{\left[ {{{\left| {{f^\prime}\left( b \right)} \right|} \over {\left| {{f^\prime}\left( x \right)} \right|}}} \right]}^{{t \over 2}}}dt} \hfill \cr {} \hfill & = \hfill & {{{\sqrt {{{\left| {{f^\prime}\left( b \right)} \right|} \over {\left| {{f^\prime}\left( x \right)} \right|}}} } \over {\ln {{\left| {{f^\prime}\left( b \right)} \right|} \over {\left| {{f^\prime}\left( x \right)} \right|}}}}B\left( {n + 1,\alpha - n} \right) - {1 \over {\ln \sqrt {{{\left| {{f^\prime}\left( b \right)} \right|} \over {\left| {{f^\prime}\left( x \right)} \right|}}} }}\int_0^1 {t^n}{{\left( {1 - t} \right)}^{\alpha - n - 1}}{{\left( {\sqrt {{{\left| {{f^\prime}\left( b \right)} \right|} \over {\left| {{f^\prime}\left( x \right)} \right|}}} } \right)}^t}dt} \hfill \cr } into Ψ1 and Ψ2 inequality and simplifying lead to the required inequality. The proof of Theorem 3 is completed.

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