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# Some new inequalities for convex functions via Riemann-Liouville fractional integrals

###### Accepted: 14 Jan 2020
Journal Details
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English

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.

#### 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: $f(m+n2)≤1n−m∫mnf(x)dx≤f(m)+f(n)2.$ 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 $Jn−αf$ 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 $Jm+αf=1Γ(α)∫mx(x−t)α−1f(t)dt, x>m$ 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 $Jn−αf=1Γ(α)∫xn(t−x)α−1f(t)dt, x 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 $Γ(α)=∫0∞e−tuα−1du$ \Gamma \left(\alpha \right) = \int_0^\infty {e^{- t}}{u^{\alpha - 1}}du . Here $Jm+0f(x)=Jn−0f(x)=f(x)$ 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 $λf(m)+μf(n)2+(2−λ−μ)2f(ξm+(1−ξ)n)−1n−m∫mnf(x)dx=n−m2[(1−ξ)∫01(2(1−ξ)t−λ)f'(t(ξm+(1−ξ)n)+(1−t)m)dt+ξ∫01(μ−2ξt)f'(t(ξm+(1−ξ)n)+(1−t)n)dt].$ \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 $λf(m)+μf(n)2+(2−λ−μ)2f(ξm+(1−ξ)n)−Γ(α+1)(n−m)α[(1−ξ)1−αJ[ξm+(1−ξ)n]−αf(m)+ξ1−αJ[ξm+(1−ξ)n]+αf(n)]=n−m2[(1−ξ)∫01(2(1−ξ)tα−λ)f'(t(ξm+(1−ξ)n)+(1−t)m)dt+ξ∫01(μ−2ξtα)f'(t(ξm+(1−ξ)n)+(1−t)n)dt]$ \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 $∫01(2(1−ξ)tα−λ)f'(t(ξm+(1−ξ)n)+(1−t)m)dt=(2(1−ξ)tα−λ)f(t(ξm+(1−ξ)n)+(1−t)m)(1−ξ)(n−m)|01 −2α(n−m)∫01tα−1f(t(ξm+(1−ξ)n)+(1−t)m)dt=[2(1−ξ)−λ]f(ξm+(1−ξ)n)+λf(m)(1−ξ)(n−m) −2α(n−m)∫01tα−1f(t(ξm+(1−ξ)n)+(1−t)m)dt=[2(1−ξ)−λ]f(ξm+(1−ξ)n)+λf(m)(1−ξ)(n−m) −2α(1−ξ)α(n−m)α+1∫mξm+(1−ξ)n(x−m)α−1f(x)dx=[2(1−ξ)−λ]f(ξm+(1−ξ)n)+λf(m)(1−ξ)(n−m) −2Γ(α+1)(1−ξ)α(n−m)α+1J[ξm+(1−ξ)n]−αf(m).$ \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 $∫01(μ−2ξtα)f'(t(ξm+(1−ξ)n)+(1−t)n)dt=(μ−2ξtα)f(t(ξm+(1−ξ)n)+(1−t)n)ξ(m−n)|01 −2α(n−m)∫01tα−1f(t(ξm+(1−ξ)n)+(1−t)n)dt=(2ξ−μ)f(ξm+(1−ξ)n)+μf(n)ξ(n−m) −2α(n−m)∫01tα−1f(t(ξm+(1−ξ)n)+(1−t)n)dt=(2ξ−μ)f(ξm+(1−ξ)n)+μf(n)ξ(n−m) −2αξα(n−m)α+1∫ξm+(1−ξ)nn(n−x)α−1f(x)dx=(2ξ−μ)f(ξm+(1−ξ)n)+μf(n)ξ(n−m) −2Γ(α+1)ξα(n−m)α+1J[ξm+(1−ξ)n]+αf(n).$ \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 $n−m2$ {{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 \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 $ρ1=λ2+αα22α(1−ξ)2α−λ2+αα22−αα(1−ξ)2α(2+α)+2(1−ξ)2+α−λ2ρ2=λ2−2(1−ξ)2+αΨ1=λ1+αα21−αα(1−ξ)1α−λ1+αα21−αα(1+α)(1−ξ)1α+2(1−ξ)1+α−λ−ρ1Ψ2=λ2−2(1−ξ)1+α+2(1−ξ)2+αγ1=μ2+αα22αξ2α−μ2+αα22−αα(2+α)ξ2α+2ξ2+α−μ2γ2=μ2−2ξ2+αδ1=μ1+αα21−ααξ1α−μ1+αα21−αα(1+α)ξ1α+2ξ1+α−μ−γ1δ2=μ2−2ξ1+α+2ξ2+α.$ \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 $|λf(m)+μf(n)2+(2−λ−μ)f(ξm+(1−ξ)n)2 −Γ(α+1)(n−m)α[(1−ξ)1−αJ[ξm+(1−ξ)n]−αf(m)+ξ1−αJ[ξm+(1−ξ)n]+αf(n)]|≤n−m2[(1−ξ)∫01|2(1−ξ)tα−λ||f'(t(ξm+(1−ξ)n)+(1−t)m)|dt +ξ∫01|μ−2ξtα||f'(t(ξm+(1−ξ)n)+(1−t)n)|dt].$ \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 $|λf(m)+μf(n)2+(2−λ−μ)f(ξm+(1−ξ)n)2 −Γ(α+1)(n−m)α[(1−ξ)1−αJ[ξm+(1−ξ)n]−αf(m)+ξ1−αJ[ξm+(1−ξ)n]+αf(n)]|≤n−m2[(1−ξ)∫01|2(1−ξ)tα−λ|[t|f'(ξm+(1−ξ)n)|+(1−t)|f'(m)|]dt +ξ∫01|μ−2ξtα|[t|f'(ξm+(1−ξ)n)|+(1−t)|f'(n)|]dt].$ \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 $∫01t|2(1−ξ)tα−λ|dt={ρ1,λ<2(1−ξ)ρ2,λ≥2(1−ξ)∫01(1−t)|2(1−ξ)tα−λ|dt={Ψ1,λ<2(1−ξ)Ψ2,λ≥2(1−ξ)∫01t|μ−2ξtα|dt={γ1,μ<2ξγ2,μ≥2ξ∫01(1−t)|μ−2ξtα|dt={δ1,μ<2ξδ2,μ≥2ξ.$ \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 \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 $ζ1=2λλ2(1−ξ)α−4(1−ξ)(α+1)(λ2(1−ξ))α+1α+2(1−ξ)α+1−λζ2=λ−2(1−ξ)α+1η1=2μμ2ξα−4ξ(α+1)(μ2ξ)α+1α+2ξα+1−μη2=μ−2ξα+1$ \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 $|λf(m)+μf(n)2+(2−λ−μ)f(ξm+(1−ξ)n)2 −Γ(α+1)(n−m)α[(1−ξ)1−αJ[ξm+(1−ξ)n]−αf(m)+ξ1−αJ[ξm+(1−ξ)n]+αf(n)]|≤n−m2[(1−ξ)∫01|2(1−ξ)tα−λ||f'(t(ξm+(1−ξ)n)+(1−t)m)|dt +ξ∫01|μ−2ξtα||f'(t(ξm+(1−ξ)n)+(1−t)n)|dt].$ \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 $|λf(m)+μf(n)2+(2−λ−μ)f(ξm+(1−ξ)n)2 −Γ(α+1)(n−m)α[(1−ξ)1−αJ[ξm+(1−ξ)n]−αf(m)+ξ1−αJ[ξm+(1−ξ)n]+αf(n)]|≤n−m2[(1−ξ)(∫01|2(1−ξ)tα−λ|dt)1−1q×(∫01|2(1−ξ)tα−λ||f'(t(ξm+(1−ξ)n)+(1−t)m)|qdt)1q +ξ(∫01|μ−2ξtα|dt)1−1q(∫01|μ−2ξtα||f'(t(ξm+(1−ξ)n)+(1−t)n)|qdt)1q].$ \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 $|λf(m)+μf(n)2+(2−λ−μ)f(ξm+(1−ξ)n)2 −Γ(α+1)(n−m)α[(1−ξ)1−αJ[ξm+(1−ξ)n]−αf(m)+ξ1−αJ[ξm+(1−ξ)n]+αf(n)]|≤n−m2[(1−ξ)(∫01|2(1−ξ)tα−λ|dt)1−1q ×(∫01|2(1−ξ)tα−λ|[t|f'(ξm+(1−ξ)n)|q+(1−t)|f'(m)|q]dt)1q+ξ(∫01|μ−2ξtα|dt)1−1q×(∫01|μ−2ξtα|[t|f'(ξm+(1−ξ)n)|q+(1−t)|f'(n)|q]dt)1q].$ \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 $∫01|2(1−ξ)tα−λ|dt={ζ1,λ<2(1−ξ)ζ2,λ≥2(1−ξ)∫01|μ−2ξtα|dt={η1,μ<2ξη2,μ≥2ξ.$ \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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• #### Exact solutions of (2 + 1)-Ablowitz-Kaup-Newell-Segur equation

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