1. bookVolume 6 (2021): Issue 1 (January 2021)
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Some new inequalities for convex functions via Riemann-Liouville fractional integrals

Published Online: 30 Jun 2020
Page range: 537 - 544
Received: 09 Jul 2019
Accepted: 14 Jan 2020
Journal Details
License
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Abstract

Fractional analysis has evolved considerably over the last decades and has become popular in many technical and scientific fields. Many integral operators which ables us to integrate from fractional orders has been generated. Each of them provides different properties such as semigroup property, singularity problems etc. In this paper, firstly, we obtained a new kernel, then some new integral inequalities which are valid for integrals of fractional orders by using Riemann-Liouville fractional integral. To do this, we used some well-known inequalities such as Hölder's inequality or power mean inequality. Our results generalize some inequalities exist in the literature.

Keywords

MSC 2010

Introduction

It is a well known fact that inequalities have important role in the studies of inequality theory, linear programming, extremum problems, optimization, error estimates and game theory. Over the years, only integer real order integrals were taken into account while handling new results about integral inequalities. However, in the recent years fractional integral operator have been considered by many scientists (see [1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]) and the references therein. There are some inequalities in the literature that accelerates studies on integral inequalities. In the following part, the Hermite-Hadamard inequality which is one of the most famous and practical inequality in the literature is given:

Theorem 1

Let f be defined from interval I (a nonempty subset of) tobe a convex function on I and m, n ∈ I with m < n. Then the double inequality given in the following holds: f(m+n2)1nmmnf(x)dxf(m)+f(n)2. f\left({{{m + n} \over 2}} \right) \le {1 \over {n - m}}\int_m^n f\left(x \right)dx \le {{f\left(m \right) + f\left(n \right)} \over 2}.

Now we will mention about Riemann-Liouville fractional integration operator (see [6]) which ables to integrate functions on fractional orders.

Definition 1

Let fL1 [m, n]. Jm+αf J_{m +}^\alpha f and Jnαf J_{n -}^\alpha f which are called left-sided and right-sided Riemann-Liouville integrals of order α > 0 with 0 ≤ mxn are defined by Jm+αf=1Γ(α)mx(xt)α1f(t)dt,x>m J_{m +}^\alpha f = {1 \over {\Gamma \left(\alpha \right)}}\int_m^x {\left({x - t} \right)^{\alpha - 1}}f\left(t \right)dt{\rm{,}}\;\;\;x > m and Jnαf=1Γ(α)xn(tx)α1f(t)dt,x<n J_{n -}^\alpha f = {1 \over {\Gamma \left(\alpha \right)}}\int_x^n {\left({t - x} \right)^{\alpha - 1}}f\left(t \right)dt{\rm{,}}\;\;\;x < n respectively where Γ(α)=0etuα1du \Gamma \left(\alpha \right) = \int_0^\infty {e^{- t}}{u^{\alpha - 1}}du . Here Jm+0f(x)=Jn0f(x)=f(x) J_{{m^ +}}^0f\left(x \right) = J_{{n^ -}}^0f\left(x \right) = f\left(x \right) .

The results have been put forward inspiring from the following kernel obtained in [9].

Lemma 2

Let f: I ⊆ ℝ → ℝ be a differentiable mapping on I° and m, nI with m < n. If f′ ∈ L1 ([m, n]), λ, μ ∈ ℝ, and ξ ∈ [0, 1], then λf(m)+μf(n)2+(2λμ)2f(ξm+(1ξ)n)1nmmnf(x)dx=nm2[(1ξ)01(2(1ξ)tλ)f'(t(ξm+(1ξ)n)+(1t)m)dt+ξ01(μ2ξt)f'(t(ξm+(1ξ)n)+(1t)n)dt]. \matrix{ {{\kern 10pt }{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)} \over 2}f\left({\xi m + \left({1 - \xi} \right)n} \right) - {1 \over {n - m}}\int_m^n f\left(x \right)dx} \hfill \cr = {{{n - m} \over 2}\left[ {\left({1 - \xi} \right)\int_0^1 \left({2\left({1 - \xi} \right)t - \lambda} \right){f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)dt} \right.} \hfill \cr {\kern 10pt }{\left. {+ \xi \int_0^1 \left({\mu - 2\xi t} \right){f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)dt} \right].} \hfill \cr}

Now firstly, we will give a new lemma including Riemann-Liouville fractional integral operator, then we will obtain new inequalities for convex functions.

Results Via Riemann-Liouville Fractional Integrals
Lemma 3

Let f : I ⊆ ℝ → ℝ be a differentiable mapping on I° and m, n ∈ I with m < n. If f′ ∈ L1 ([m, n]), λ, μ ∈ ℝ, α > 0 and ξ ∈ [0, 1] then λf(m)+μf(n)2+(2λμ)2f(ξm+(1ξ)n)Γ(α+1)(nm)α[(1ξ)1αJ[ξm+(1ξ)n]αf(m)+ξ1αJ[ξm+(1ξ)n]+αf(n)]=nm2[(1ξ)01(2(1ξ)tαλ)f'(t(ξm+(1ξ)n)+(1t)m)dt+ξ01(μ2ξtα)f'(t(ξm+(1ξ)n)+(1t)n)dt] \matrix{{\kern 10pt}{{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)} \over 2}f\left({\xi m + \left({1 - \xi} \right)n} \right)} \hfill \cr {\kern 10pt} {- {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \hfill \cr = {{{n - m} \over 2}\left[ {\left({1 - \xi} \right)\int_0^1 \left({2\left({1 - \xi} \right){t^\alpha} - \lambda} \right){f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)dt} \right.} \hfill \cr {\kern 10pt} {\left. {+ \xi \int_0^1 \left({\mu - 2\xi {t^\alpha}} \right){f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)dt} \right]} \hfill} where Γ(.) is the gamma function.

Proof

Integrating by part and changing variables of integration x = t (ξ m + (1 − ξ) n) + (1 − t) m yield 01(2(1ξ)tαλ)f'(t(ξm+(1ξ)n)+(1t)m)dt=(2(1ξ)tαλ)f(t(ξm+(1ξ)n)+(1t)m)(1ξ)(nm)|012α(nm)01tα1f(t(ξm+(1ξ)n)+(1t)m)dt=[2(1ξ)λ]f(ξm+(1ξ)n)+λf(m)(1ξ)(nm)2α(nm)01tα1f(t(ξm+(1ξ)n)+(1t)m)dt=[2(1ξ)λ]f(ξm+(1ξ)n)+λf(m)(1ξ)(nm)2α(1ξ)α(nm)α+1mξm+(1ξ)n(xm)α1f(x)dx=[2(1ξ)λ]f(ξm+(1ξ)n)+λf(m)(1ξ)(nm)2Γ(α+1)(1ξ)α(nm)α+1J[ξm+(1ξ)n]αf(m). \matrix{{\,\,\,\,\,\int_0^1 \left({2\left({1 - \xi} \right){t^\alpha} - \lambda} \right){f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)dt} \hfill \cr {= \left. {\left({2\left({1 - \xi} \right){t^\alpha} - \lambda} \right){{f\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)} \over {\left({1 - \xi} \right)\left({n - m} \right)}}} \right|_0^1} \hfill \cr {\,\,\,\,\, - {{2\alpha} \over {\left({n - m} \right)}}\int_0^1 {t^{\alpha - 1}}f\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)dt} \hfill \cr {= {{\left[ {2\left({1 - \xi} \right) - \lambda} \right]f\left({\xi m + \left({1 - \xi} \right)n} \right) + \lambda f\left(m \right)} \over {\left({1 - \xi} \right)\left({n - m} \right)}}} \hfill \cr {\,\,\,\,\, - {{2\alpha} \over {\left({n - m} \right)}}\int_0^1 {t^{\alpha - 1}}f\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)dt} \hfill \cr {= {{\left[ {2\left({1 - \xi} \right) - \lambda} \right]f\left({\xi m + \left({1 - \xi} \right)n} \right) + \lambda f\left(m \right)} \over {\left({1 - \xi} \right)\left({n - m} \right)}}} \hfill \cr {\,\,\,\,\, - {{2\alpha} \over {{{\left({1 - \xi} \right)}^\alpha}{{\left({n - m} \right)}^{\alpha + 1}}}}\int_m^{\xi m + \left({1 - \xi} \right)n} {{\left({x - m} \right)}^{\alpha - 1}}f\left(x \right)dx} \hfill \cr {= {{\left[ {2\left({1 - \xi} \right) - \lambda} \right]f\left({\xi m + \left({1 - \xi} \right)n} \right) + \lambda f\left(m \right)} \over {\left({1 - \xi} \right)\left({n - m} \right)}}} \hfill \cr {\,\,\,\,\, - {{2\Gamma \left({\alpha + 1} \right)} \over {{{\left({1 - \xi} \right)}^\alpha}{{\left({n - m} \right)}^{\alpha + 1}}}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right).} \hfill \cr} On the other hand, with similar way and changing variables of integration x = t (ξ m + (1 − ξ) n)+(1 − t) n yield 01(μ2ξtα)f'(t(ξm+(1ξ)n)+(1t)n)dt=(μ2ξtα)f(t(ξm+(1ξ)n)+(1t)n)ξ(mn)|012α(nm)01tα1f(t(ξm+(1ξ)n)+(1t)n)dt=(2ξμ)f(ξm+(1ξ)n)+μf(n)ξ(nm)2α(nm)01tα1f(t(ξm+(1ξ)n)+(1t)n)dt=(2ξμ)f(ξm+(1ξ)n)+μf(n)ξ(nm)2αξα(nm)α+1ξm+(1ξ)nn(nx)α1f(x)dx=(2ξμ)f(ξm+(1ξ)n)+μf(n)ξ(nm)2Γ(α+1)ξα(nm)α+1J[ξm+(1ξ)n]+αf(n). \matrix{{\,\,\,\,\,\int_0^1 \left({\mu - 2\xi {t^\alpha}} \right){f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)dt} \hfill \cr {= \left. {\left({\mu - 2\xi {t^\alpha}} \right){{f\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)} \over {\xi \left({m - n} \right)}}} \right|_0^1} \hfill \cr {\,\,\,\,\, - {{2\alpha} \over {\left({n - m} \right)}}\int_0^1 {t^{\alpha - 1}}f\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)dt} \hfill \cr {= {{\left({2\xi - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right) + \mu f\left(n \right)} \over {\xi \left({n - m} \right)}}} \hfill \cr {\,\,\,\,\, - {{2\alpha} \over {\left({n - m} \right)}}\int_0^1 {t^{\alpha - 1}}f\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)dt} \hfill \cr {= {{\left({2\xi - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right) + \mu f\left(n \right)} \over {\xi \left({n - m} \right)}}} \hfill \cr {\,\,\,\,\, - {{2\alpha} \over {{\xi ^\alpha}{{\left({n - m} \right)}^{\alpha + 1}}}}\int_{\xi m + \left({1 - \xi} \right)n}^n {{\left({n - x} \right)}^{\alpha - 1}}f\left(x \right)dx} \hfill \cr {= {{\left({2\xi - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right) + \mu f\left(n \right)} \over {\xi \left({n - m} \right)}}} \hfill \cr {\,\,\,\,\, - {{2\Gamma \left({\alpha + 1} \right)} \over {{\xi ^\alpha}{{\left({n - m} \right)}^{\alpha + 1}}}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right).} \hfill \cr}

By multiplying 4 with (1 − ξ), 5 with ξ, summing them side by side and multiplying the equality with nm2 {{n - m} \over 2} we get the desired result.

Remark 1

If we choose α = 1 in Lemma 3, we gel Lemma 2.

Theorem 4

Let f : I ⊆ ℝ → ℝ be a differentiable mapping on I° and m, nI with m < n. If |f′| is convex on I, f′L1 ([m, n]), λ, μ ∈ ℝ+, α > 0 and ξ ∈ [0, 1] then |λf(m)+μf(n)2+(2λμ)f(ξm+(1ξ)n)2Γ(α+1)(nm)α[(1ξ)1αJ[ξm+(1ξ)n]αf(m)+ξ1αJ[ξm+(1ξ)n]+αf(n)]|nm2{(1ξ)[ρ1|f'(ξm+(1ξ)n)|+Ψ1|f'(m)|]+ξ[γ1|f'(ξm+(1ξ)n)|+δ1|f'(m)|] for λ<2(1ξ) and μ<2ξ(1ξ)[ρ1|f'(ξm+(1ξ)n)|+Ψ1|f'(m)|]+ξ[γ2|f'(ξm+(1ξ)n)|+δ2|f'(m)|] for λ<2(1ξ) and μ2ξ(1ξ)[ρ2|f'(ξm+(1ξ)n)|+Ψ2|f'(m)|]+ξ[γ1|f'(ξm+(1ξ)n)|+δ1|f'(m)|] for λ2(1ξ) and μ<2ξ(1ξ)[ρ2|f'(ξm+(1ξ)n)|+Ψ2|f'(m)|]+ξ[γ2|f'(ξm+(1ξ)n)|+δ2|f'(m)|] for λ2(1ξ) and μ2ξ \matrix{{\,\,\,\,\,\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {\,\,\,\, - {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left\{{\matrix{{\matrix{{\left({1 - \xi} \right)\left[ {{\rho _1}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\Psi _1}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr {+ \xi \left[ {{\gamma _1}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\delta _1}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr}} & {for\,\,\,\lambda < 2\left({1 - \xi} \right)\,\,\,and\,\,\,\mu < 2\xi} \cr {} & {} \cr {\matrix{{\left({1 - \xi} \right)\left[ {{\rho _1}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\Psi _1}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr {+ \xi \left[ {{\gamma _2}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\delta _2}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr}} & {for\,\,\,\lambda < 2\left({1 - \xi} \right)\,\,\,and\,\,\,\mu \ge 2\xi} \cr {} & {} \cr {\matrix{{\left({1 - \xi} \right)\left[ {{\rho _2}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\Psi _2}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr {+ \xi \left[ {{\gamma _1}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\delta _1}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr}} & {for\,\,\,\lambda \ge 2\left({1 - \xi} \right)\,\,\,and\,\,\,\mu < 2\xi} \cr {} & {} \cr {\matrix{{\left({1 - \xi} \right)\left[ {{\rho _2}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\Psi _2}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr {+ \xi \left[ {{\gamma _2}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\delta _2}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr}} & {for\,\,\,\lambda \ge 2\left({1 - \xi} \right)\,\,\,and\,\,\,\mu \ge 2\xi} \cr}} \right.} \hfill \cr} where Γ(.) is the gamma function and ρ1=λ2+αα22α(1ξ)2αλ2+αα22αα(1ξ)2α(2+α)+2(1ξ)2+αλ2ρ2=λ22(1ξ)2+αΨ1=λ1+αα21αα(1ξ)1αλ1+αα21αα(1+α)(1ξ)1α+2(1ξ)1+αλρ1Ψ2=λ22(1ξ)1+α+2(1ξ)2+αγ1=μ2+αα22αξ2αμ2+αα22αα(2+α)ξ2α+2ξ2+αμ2γ2=μ22ξ2+αδ1=μ1+αα21ααξ1αμ1+αα21αα(1+α)ξ1α+2ξ1+αμγ1δ2=μ22ξ1+α+2ξ2+α. \matrix{{{\rho _1} = {{{\lambda ^{{{2 + \alpha} \over \alpha}}}} \over {{2^{{2 \over \alpha}}}{{\left({1 - \xi} \right)}^{{2 \over \alpha}}}}} - {{{\lambda ^{{{2 + \alpha} \over \alpha}}}} \over {{2^{{{2 - \alpha} \over \alpha}}}{{\left({1 - \xi} \right)}^{{2 \over \alpha}}}\left({2 + \alpha} \right)}} + {{2\left({1 - \xi} \right)} \over {2 + \alpha}} - {\lambda \over 2}} \hfill \cr {{\rho _2} = {\lambda \over 2} - {{2\left({1 - \xi} \right)} \over {2 + \alpha}}} \hfill \cr {{\Psi _1} = {{{\lambda ^{{{1 + \alpha} \over \alpha}}}} \over {{2^{{{1 - \alpha} \over \alpha}}}{{\left({1 - \xi} \right)}^{{1 \over \alpha}}}}} - {{{\lambda ^{{{1 + \alpha} \over \alpha}}}} \over {{2^{{{1 - \alpha} \over \alpha}}}\left({1 + \alpha} \right){{\left({1 - \xi} \right)}^{{1 \over \alpha}}}}} + {{2\left({1 - \xi} \right)} \over {1 + \alpha}} - \lambda - {\rho _1}} \hfill \cr {{\Psi _2} = {\lambda \over 2} - {{2\left({1 - \xi} \right)} \over {1 + \alpha}} + {{2\left({1 - \xi} \right)} \over {2 + \alpha}}} \hfill \cr {{\gamma _1} = {{{\mu ^{{{2 + \alpha} \over \alpha}}}} \over {{2^{{2 \over \alpha}}}{\xi ^{{2 \over \alpha}}}}} - {{{\mu ^{{{2 + \alpha} \over \alpha}}}} \over {{2^{{{2 - \alpha} \over \alpha}}}\left({2 + \alpha} \right){\xi ^{{2 \over \alpha}}}}} + {{2\xi} \over {2 + \alpha}} - {\mu \over 2}} \hfill \cr {{\gamma _2} = {\mu \over 2} - {{2\xi} \over {2 + \alpha}}} \hfill \cr {{\delta _1} = {{{\mu ^{{{1 + \alpha} \over \alpha}}}} \over {{2^{{{1 - \alpha} \over \alpha}}}{\xi ^{{1 \over \alpha}}}}} - {{{\mu ^{{{1 + \alpha} \over \alpha}}}} \over {{2^{{{1 - \alpha} \over \alpha}}}\left({1 + \alpha} \right){\xi ^{{1 \over \alpha}}}}} + {{2\xi} \over {1 + \alpha}} - \mu - {\gamma _1}} \hfill \cr {{\delta _2} = {\mu \over 2} - {{2\xi} \over {1 + \alpha}} + {{2\xi} \over {2 + \alpha}}.} \hfill \cr}

Proof

By using Lemma 3 and using properties of absolute value we have |λf(m)+μf(n)2+(2λμ)f(ξm+(1ξ)n)2Γ(α+1)(nm)α[(1ξ)1αJ[ξm+(1ξ)n]αf(m)+ξ1αJ[ξm+(1ξ)n]+αf(n)]|nm2[(1ξ)01|2(1ξ)tαλ||f'(t(ξm+(1ξ)n)+(1t)m)|dt+ξ01|μ2ξtα||f'(t(ξm+(1ξ)n)+(1t)n)|dt]. \matrix{{\,\,\,\,\,\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {\,\,\,\,\, - {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left[ {\left({1 - \xi} \right)\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|\left| {{f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)} \right|dt} \right.} \hfill \cr {\left. {\,\,\,\,\, + \xi \int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|\left| {{f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)} \right|dt} \right].} \hfill \cr}

Using convexity of |f′| we have |λf(m)+μf(n)2+(2λμ)f(ξm+(1ξ)n)2Γ(α+1)(nm)α[(1ξ)1αJ[ξm+(1ξ)n]αf(m)+ξ1αJ[ξm+(1ξ)n]+αf(n)]|nm2[(1ξ)01|2(1ξ)tαλ|[t|f'(ξm+(1ξ)n)|+(1t)|f'(m)|]dt+ξ01|μ2ξtα|[t|f'(ξm+(1ξ)n)|+(1t)|f'(n)|]dt]. \matrix{{\,\,\,\,\,\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {\,\,\,\, - {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left[ {\left({1 - \xi} \right)\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|\left[ {t\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + \left({1 - t} \right)\left| {{f^{'}}\left(m \right)} \right|} \right]dt} \right.} \hfill \cr {\left. {\,\,\,\, + \xi \int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|\left[ {t\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + \left({1 - t} \right)\left| {{f^{'}}\left(n \right)} \right|} \right]dt} \right].} \hfill \cr}

With simple calculations it can be seen 01t|2(1ξ)tαλ|dt={ρ1,λ<2(1ξ)ρ2,λ2(1ξ)01(1t)|2(1ξ)tαλ|dt={Ψ1,λ<2(1ξ)Ψ2,λ2(1ξ)01t|μ2ξtα|dt={γ1,μ<2ξγ2,μ2ξ01(1t)|μ2ξtα|dt={δ1,μ<2ξδ2,μ2ξ. \matrix{{\kern 32pt}{\int_0^1 t\left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|dt =} \hfill & {\left\{{\matrix{{{\rho _1},} & {\lambda < 2\left({1 - \xi} \right)} \cr {{\rho _2},} & {\lambda \ge 2\left({1 - \xi} \right)} \cr}} \right.} \hfill \cr {\int_0^1 \left({1 - t} \right)\left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|dt =} \hfill & {\left\{{\matrix{{{\Psi _1},} & {\lambda < 2\left({1 - \xi} \right)} \cr {{\Psi _2},} & {\lambda \ge 2\left({1 - \xi} \right)} \cr}} \right.} \hfill \cr {\kern 65pt}{\int_0^1 t\left| {\mu - 2\xi {t^\alpha}} \right|dt =} \hfill & {\left\{{\matrix{{{\gamma _1},} & {\mu < 2\xi} \cr {{\gamma _2},} & {\mu \ge 2\xi} \cr}} \right.} \hfill \cr{\kern 32pt} {\int_0^1 \left({1 - t} \right)\left| {\mu - 2\xi {t^\alpha}} \right|dt =} \hfill & {\left\{{\matrix{{{\delta _1},} & {\mu < 2\xi} \cr {{\delta _2},} & {\mu \ge 2\xi} \cr}} \right..} \hfill \cr}

By using necessary coefficients in (6), the proof is completed.

Theorem 5

Let f: I ⊆ ℝ → ℝ be a differentiable mapping on I° and m, n ∈ I with m < n. If |f′|q is convex on I, f′ ∈ L1 ([m, n]), λ, μ ∈ ℝ+, α > 0, ξ ∈ [0, 1] then |λf(m)+μf(n)2+(2λμ)f(ξm+(1ξ)n)2Γ(α+1)(nm)α[(1ξ)1αJ[ξm+(1ξ)n]αf(m)+ξ1αJ[ξm+(1ξ)n]+αf(n)]|nm2{(1ξ)(ζ1)11q[ρ1|f'(ξm+(1ξ)n)|q+Ψ1|f'(m)|q]1q+ξ(η1)11q[γ1|f'(ξm+(1ξ)n)|q+δ1|f'(n)|q]1q, for λ<2(1ξ) and μ<2ξ;(1ξ)(ζ1)11q[ρ1|f'(ξm+(1ξ)n)|q+Ψ1|f'(m)|q]1q+ξ(η2)11q[γ2|f'(ξm+(1ξ)n)|q+δ2|f'(n)|q]1q, for λ<2(1ξ) and μ2ξ;(1ξ)(ζ2)11q[ρ2|f'(ξm+(1ξ)n)|q+Ψ2|f'(m)|q]1q+ξ(η1)11q[γ1|f'(ξm+(1ξ)n)|q+δ1|f'(n)|q]1q, for λ2(1ξ) and μ<2ξ;(1ξ)(ζ2)11q[ρ2|f'(ξm+(1ξ)n)|q+Ψ2|f'(m)|q]1q+ξ(η2)11q[γ2|f'(ξm+(1ξ)n)|q+δ2|f'(n)|q]1q, for λ2(1ξ) and μ2ξ \matrix{{\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {- {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left\{{\matrix{{\matrix{{\left({1 - \xi} \right){{\left({{\zeta _1}} \right)}^{1 - {1 \over q}}}{{\left[ {{\rho _1}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\Psi _1}{{\left| {{f^{'}}\left(m \right)} \right|}^q}} \right]}^{{1 \over q}}}} \cr {+ \xi {{\left({{\eta _1}} \right)}^{1 - {1 \over q}}}{{\left[ {{\gamma _1}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\delta _1}{{\left| {{f^{'}}\left(n \right)} \right|}^q}} \right]}^{{1 \over q}}},} \cr {{\rm{for}}\lambda < 2\left({1 - \xi} \right){\rm{and}}\mu < 2\xi ;} \cr}} \cr {} \cr {\matrix{{\left({1 - \xi} \right){{\left({{\zeta _1}} \right)}^{1 - {1 \over q}}}{{\left[ {{\rho _1}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\Psi _1}{{\left| {{f^{'}}\left(m \right)} \right|}^q}} \right]}^{{1 \over q}}}} \cr {+ \xi {{\left({{\eta _2}} \right)}^{1 - {1 \over q}}}{{\left[ {{\gamma _2}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\delta _2}{{\left| {{f^{'}}\left(n \right)} \right|}^q}} \right]}^{{1 \over q}}},} \cr {{\rm{for}}\lambda < 2\left({1 - \xi} \right){\rm{and}}\mu \ge 2\xi ;} \cr}} \cr {} \cr {\matrix{{\left({1 - \xi} \right){{\left({{\zeta _2}} \right)}^{1 - {1 \over q}}}{{\left[ {{\rho _2}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\Psi _2}{{\left| {{f^{'}}\left(m \right)} \right|}^q}} \right]}^{{1 \over q}}}} \cr {+ \xi {{\left({{\eta _1}} \right)}^{1 - {1 \over q}}}{{\left[ {{\gamma _1}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\delta _1}{{\left| {{f^{'}}\left(n \right)} \right|}^q}} \right]}^{{1 \over q}}},} \cr {{\rm{for}}\lambda \ge 2\left({1 - \xi} \right){\rm{and}}\mu < 2\xi ;} \cr}} \cr {} \cr {\matrix{{\left({1 - \xi} \right){{\left({{\zeta _2}} \right)}^{1 - {1 \over q}}}{{\left[ {{\rho _2}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\Psi _2}{{\left| {{f^{'}}\left(m \right)} \right|}^q}} \right]}^{{1 \over q}}}} \cr {+ \xi {{\left({{\eta _2}} \right)}^{1 - {1 \over q}}}{{\left[ {{\gamma _2}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\delta _2}{{\left| {{f^{'}}\left(n \right)} \right|}^q}} \right]}^{{1 \over q}}},} \cr {{\rm{for}}\lambda \ge 2\left({1 - \xi} \right){\rm{and}}\mu \ge 2\xi} \cr}} \cr}} \right.} \hfill \cr} where q ≥ 1, Γ(.) is the gamma function and ζ1=2λλ2(1ξ)α4(1ξ)(α+1)(λ2(1ξ))α+1α+2(1ξ)α+1λζ2=λ2(1ξ)α+1η1=2μμ2ξα4ξ(α+1)(μ2ξ)α+1α+2ξα+1μη2=μ2ξα+1 \matrix{{{\zeta _1} = 2\lambda \root \alpha \of {{\lambda \over {2\left({1 - \xi} \right)}}} - {{4\left({1 - \xi} \right)} \over {\left({\alpha + 1} \right)}}\root \alpha \of {{{\left({{\lambda \over {2\left({1 - \xi} \right)}}} \right)}^{\alpha + 1}}} + {{2\left({1 - \xi} \right)} \over {\alpha + 1}} - \lambda} \hfill \cr {{\zeta _2} = \lambda - {{2\left({1 - \xi} \right)} \over {\alpha + 1}}} \hfill \cr {{\eta _1} = 2\mu \root \alpha \of {{\mu \over {2\xi}}} - {{4\xi} \over {\left({\alpha + 1} \right)}}\root \alpha \of {{{\left({{\mu \over {2\xi}}} \right)}^{\alpha + 1}}} + {{2\xi} \over {\alpha + 1}} - \mu} \hfill \cr {{\eta _2} = \mu - {{2\xi} \over {\alpha + 1}}} \hfill \cr} with ρ1, ρ2, Ψ1, Ψ2, γ1, γ2, δ1, δ2 described as in Theorem 4.

Proof

By using Lemma 3 and using properties of absolute value we have |λf(m)+μf(n)2+(2λμ)f(ξm+(1ξ)n)2Γ(α+1)(nm)α[(1ξ)1αJ[ξm+(1ξ)n]αf(m)+ξ1αJ[ξm+(1ξ)n]+αf(n)]|nm2[(1ξ)01|2(1ξ)tαλ||f'(t(ξm+(1ξ)n)+(1t)m)|dt+ξ01|μ2ξtα||f'(t(ξm+(1ξ)n)+(1t)n)|dt]. \matrix{{\,\,\,\,\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {\,\,\, - {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left[ {\left({1 - \xi} \right)\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|\left| {{f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)} \right|dt} \right.} \hfill \cr {\left. {\,\,\, + \xi \int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|\left| {{f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)} \right|dt} \right].} \hfill \cr}

Using power mean inequality, it yields |λf(m)+μf(n)2+(2λμ)f(ξm+(1ξ)n)2Γ(α+1)(nm)α[(1ξ)1αJ[ξm+(1ξ)n]αf(m)+ξ1αJ[ξm+(1ξ)n]+αf(n)]|nm2[(1ξ)(01|2(1ξ)tαλ|dt)11q×(01|2(1ξ)tαλ||f'(t(ξm+(1ξ)n)+(1t)m)|qdt)1q+ξ(01|μ2ξtα|dt)11q(01|μ2ξtα||f'(t(ξm+(1ξ)n)+(1t)n)|qdt)1q]. \matrix{{\,\,\,\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {\,\,\, - {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left[ {\matrix{{\left({1 - \xi} \right){{\left({\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|dt} \right)}^{1 - {1 \over q}}}} \cr {\times {{\left({\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|{{\left| {{f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)} \right|}^q}dt} \right)}^{{1 \over q}}}} \cr}} \right.} \hfill \cr {\left. {\,\,\, + \xi {{\left({\int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|dt} \right)}^{1 - {1 \over q}}}{{\left({\int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|{{\left| {{f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)} \right|}^q}dt} \right)}^{{1 \over q}}}} \right].} \hfill \cr}

By taking into account convexity of |f′|q we get |λf(m)+μf(n)2+(2λμ)f(ξm+(1ξ)n)2Γ(α+1)(nm)α[(1ξ)1αJ[ξm+(1ξ)n]αf(m)+ξ1αJ[ξm+(1ξ)n]+αf(n)]|nm2[(1ξ)(01|2(1ξ)tαλ|dt)11q×(01|2(1ξ)tαλ|[t|f'(ξm+(1ξ)n)|q+(1t)|f'(m)|q]dt)1q+ξ(01|μ2ξtα|dt)11q×(01|μ2ξtα|[t|f'(ξm+(1ξ)n)|q+(1t)|f'(n)|q]dt)1q]. \matrix{{\,\,\,\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {\,\,\, - {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left[ {\matrix{{\left({1 - \xi} \right){{\left({\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|dt} \right)}^{1 - {1 \over q}}}} \cr {\times {{\left({\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|\left[ {t{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + \left({1 - t} \right){{\left| {{f^{'}}\left(m \right)} \right|}^q}} \right]dt} \right)}^{{1 \over q}}}} \cr}} \right.} \hfill \cr {\left. {\matrix{{+ \xi {{\left({\int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|dt} \right)}^{1 - {1 \over q}}}} \cr {\,\,\,\times {{\left({\int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|\left[ {t{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + \left({1 - t} \right){{\left| {{f^{'}}\left(n \right)} \right|}^q}} \right]dt} \right)}^{{1 \over q}}}} \cr}} \right].} \hfill \cr}

By making necessary computations we have 01|2(1ξ)tαλ|dt={ζ1,λ<2(1ξ)ζ2,λ2(1ξ)01|μ2ξtα|dt={η1,μ<2ξη2,μ2ξ. \matrix{{\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|dt =} \hfill & {\left\{{\matrix{{{\zeta _1},} & {\lambda < 2\left({1 - \xi} \right)} \cr {{\zeta _2},} & {\lambda \ge 2\left({1 - \xi} \right)} \cr}} \right.} \hfill \cr {{\kern 30pt}\int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|dt =} \hfill & {\left\{{\matrix{{{\eta _1},} & {\mu < 2\xi} \cr {{\eta _2},} & {\mu \ge 2\xi} \cr}} \right..} \hfill \cr}

Conclusions

A new Lemma was proved in this study. Using this lemma, new fractional type inequalities were obtained. New theorems for different types of convex functions can be obtained by using Lemma 3, and thus, new upper bounds can be obtained. Various applications for these inequalities can be revealed. Also Lemma 2 can be generalized and new integral inequalities can be obtained through different fractional integral operators.

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