On Some Integral Inequalities via Conformable Fractional Integrals

[See [5]] A mapping f : I → [0, ∞) is said to be log−convex or multiplicatively convex if log f is convex or equivalently for all θ, υ ∈ I and ς ∈ [0, 1], one has the inequality: (1.1) f(ςθ+(1−ς)ϑ)≤[f(θ)]ς[f(ϑ)]1−ς f\left( {\varsigma \theta + (1 - \varsigma )\vartheta } \right) \le {\left[ {f(\theta )} \right]^\varsigma }{\left[ {f(\vartheta )} \right]^{1 - \varsigma }}

[See [6]] Let s ∈ (0, 1]. A function f : [0, ∞) → [0, ∞) is said to be s−convex in the second sense if (1.2) f(tx+(1−t)y)≤tsf(x)+(1−t)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].

(See [5]) A function f : [a,b] → ℝ is said quasi-convex on [a,b] if f(λx+(1−λ)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].

Let f ∈ L₁[a,b]. The Riemann-Liouville integrals Ja+αf J_{a + }^\alpha f and Jb−αf J_{b - }^\alpha f of order α > 0 are defined by Ja+αf(t)=1Γ(α)∫at(t−x)α−1f(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 Jb−αf(t)=1Γ(α)∫tb(x−t)α−1f(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)=∫0∞e−ttα−1dt \Gamma (\alpha ) = \int_0^\infty {e^{ - t}}{t^{\alpha - 1}}dt . Here Ja+0f(t)=Jb−0f(t)=f(t) J_{a + }^0f(t) = J_{b - }^0f(t) = f(t)

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

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

(See [3]) Let f : [a,b] ⊂ ℝ → ℝ be a differentiable mapping on (a, b) such that f^′ ∈ L₁ [a, b]. Then for all x ∈ [a, b] and α ∈ (n, n + 1], the following equality holds: (x−a)α+1n!(b−a)[∫01Bt(n+1,α−n)2f'(1+t2x+1−t2a)dt−∫01Bt(n+1,α−n)2f'(1−t2x+1+t2a)dt]−(b−x)α+1n!(b−a)[∫01Bt(n+1,α−n)2f'(1+t2x+1−t2b)dt+∫01Bt(n+1,α−n)2f'(1−t2x+1+t2b)dt]=Γ(α−n)(x−a)αΓ(α+1)[f(x)+f(a)](b−a)−2αb−a[xIαf(x+a2)+Iαaf(x+a2)]+Γ(α−n)(b−x)αΓ(α+1)[f(x)+f(b)](b−a)−2αb−a[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 B_t (a,b) is incompleted beta function.

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.

Let f : [a, b] ⊂ ℝ → ℝ be a differentiable mapping on (a, b) such that f^′ ∈ L₁ [a, b]. If | f^′ | is quasi-convex on [a, b], then the following inequality holds for conformable fractional integrals: |Ff(α,n;x)|≤Γ(α−n+1)Γ(α+2)(b−a)((x−a)α+1sup{|f'(x)|,|f'(a)|}+(b−x)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...

Since | f^′ | is quasi-convex on [a, b] and by Lemma 1, we can write I1=|(x−a)α+1n!(b−a)[∫01Bt(n+1,α−n)2f'(1+t2x+1−t2a)dt−∫01Bt(n+1,α−n)2f'(1−t2x+1+t2a)dt]|≤(x−a)α+1n!(b−a)[∫01|Bt(n+1,α−n)2||f'(1+t2x+1−t2a)|dt+∫01|Bt(n+1,α−n)2||f'(1−t2x+1+t2a)|dt]≤(x−a)α+1n!(b−a)sup{|f'(x)|,|f'(a)|}∫01|Bt(n+1,α−n)|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=|(b−x)α+1n!(b−a)[∫01Bt(n+1,α−n)2f'(1+t2x+1−t2b)dt−∫01Bt(n+1,α−n)2f'(1−t2x+1+t2b)dt]|≤(b−x)α+1n!(b−a)sup{|f'(x)|,|f'(b)|}∫01|Bt(n+1,α−n)|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 }

Under conditions of Theorem 1, if α = n + 1, then |(x−a)n+1Γ(n+2)(b−a)[f'(x)+f'(a)]−2n+1(b−a)[Jx−n+1+Ja+n+1]+(b−x)n+1Γ(n+2)(b−a)[f'(x)+f'(b)]−2n+1(b−a)[Jx−n+1+Jb+n+1]|≤(x−a)n+2sup{|f'(x)|,|f'(a)|}+(b−x)n+2sup{|f'(x)|,|f'(b)|}(n+2)!(b−a) \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 }

Let f : [a, b] ⊂ ℝ → ℝ be a differentiable mapping on (a, b) such that f^′ ∈ L₁ [a, b]. If | f^′ | is s−convex 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!(b−a)[(x−a)α+1(|f'(x)|+|f'(a)|)+(b−x)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,α−n)|(1+t)sdt=Γ(n+1)Γ(α−n)×[2s−Γ(α+1)2F1(n+1,−1−s;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,α−n)|(1−t)sdt=B(n+1,α−n+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 }

Since |f^′ | is s−convex on [a, b] and by Lemma 1, we can write Q1=|(x−a)α+1n!(b−a)[∫01Bt(n+1,α−n)2f'(1+t2x+1−t2a)dt−∫01Bt(n+1,α−n)2f'(1−t2x+1+t2a)dt]|≤(x−a)α+1n!(b−a)[∫01|Bt(n+1,α−n)2||f'(1+t2x+1−t2a)|dt+∫01|Bt(n+1,α−n)2||f'(1−t2x+1+t2a)|dt]≤(x−a)α+12n!(b−a)[∫01|Bt(n+1,α−n)|{(1+t2)s|f'(x)|+(1−t2)s|f'(a)|}dt+∫01|Bt(n+1,α−n)|{(1−t2)s|f'(x)|+(1+t2)s|f'(a)|}dt]=(x−a)α+12s+1n!(b−a)[|f'(x)|∫01|Bt(n+1,α−n)|(1+t)sdt+|f'(a)|∫01|Bt(n+1,α−n)|(1−t)sdt+|f'(x)|∫01|Bt(n+1,α−n)|(1−t)sdt+|f'(a)|∫01|Bt(n+1,α−n)|(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=|(b−x)α+1n!(b−a)[∫01Bt(n+1,α−n)2f'(1+t2x+1−t2b)dt−∫01Bt(n+1,α−n)2f'(1−t2x+1+t2b)dt]|≤(b−x)α+12s+1n!(b−a)[|f'(x)|∫01|Bt(n+1,α−n)|(1+t)sdt+|f'(b)|∫01|Bt(n+1,α−n)|(1−t)sdt+|f'(x)|∫01|Bt(n+1,α−n)|(1−t)sdt+|f'(b)|∫01|Bt(n+1,α−n)|(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.

Under conditions of Theorem 2,

if s = 1, then |Ff(α,n;x)|≤14n!(b−a)[2B(n+1,α−n)−B(n+1,α−n−1)−B(n+2,α−n)−12B(n+3,α−n)]×[(x−a)α+1(|f'(x)|+|f'(a)|)+(b−x)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 }

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,α−n)|f'(x)|122n!(b−a)[(x−a)α+1|f'(a)|12(Ω1+Ω2)+(b−x)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...

Since | f^′ | is log−convex on [a, b] and by Lemma 1, we can write Ψ1=|(x−a)α+1n!(b−a)[∫01Bt(n+1,α−n)2f'(1+t2x+1−t2a)dt−∫01Bt(n+1,α−n)2f'(1−t2x+1+t2a)dt]|≤(x−a)α+1n!(b−a)[∫01|Bt(n+1,α−n)2||f'(1+t2x+1−t2a)|dt+∫01|Bt(n+1,α−n)2||f'(1−t2x+1+t2a)|dt]≤(x−a)α+12n!(b−a)[∫01|Bt(n+1,α−n)||f'(x)|(1+t2)|f'(a)|(1−t2)dt+∫01|Bt(n+1,α−n)||f'(x)|(1−t2)|f'(a)|(1+t2)dt]=(x−a)α+12n!(b−a)|f'(x)|12|f'(a)|12[∫01|Bt(n+1,α−n)|[|f'(x)||f'(a)|]t2dt+∫01|Bt(n+1,α−n)|[|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=|(b−x)α+1n!(b−a)[∫01Bt(n+1,α−n)2f'(1+t2x+1−t2b)dt−∫01Bt(n+1,α−n)2f'(1−t2x+1+t2b)dt]|≤(b−x)α+1n!(b−a)[∫01|Bt(n+1,α−n)2||f'(1+t2x+1−t2b)|dt+∫01|Bt(n+1,α−n)2||f'(1−t2x+1+t2b)|dt]≤(b−x)α+12n!(b−a)|f'(x)|12|f'(b)|12[∫01|Bt(n+1,α−n)|[|f'(x)||f'(b)|]t2dt+∫01|Bt(n+1,α−n)|[|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,α−n)|[|f'(x)||f'(a)|]t2dt=|f'(x)||f'(a)|ln|f'(x)||f'(a)|B(n+1,α−n)−1ln|f'(x)||f'(a)|∫01tn(1−t)α−n−1(|f'(x)||f'(a)|)tdtΩ2=∫01|Bt(n+1,α−n)|[|f'(a)||f'(x)|]t2dt=|f'(a)||f'(x)|ln|f'(a)||f'(x)|B(n+1,α−n)−1ln|f'(a)||f'(x)|∫01tn(1−t)α−n−1(|f'(a)||f'(x)|)tdtΩ3=∫01|Bt(n+1,α−n)|[|f'(x)||f'(b)|]t2dt=|f'(x)||f'(b)|ln|f'(x)||f'(b)|B(n+1,α−n)−1ln|f'(x)||f'(b)|∫01tn(1−t)α−n−1(|f'(x)||f'(b)|)tdtΩ4=∫01|Bt(n+1,α−n)|[|f'(b)||f'(x)|]t2dt=|f'(b)||f'(x)|ln|f'(b)||f'(x)|B(n+1,α−n)−1ln|f'(b)||f'(x)|∫01tn(1−t)α−n−1(|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.