# Some Ostrowski Type Inequalities for Harmonicallys,m-Convex Functions in Second Sense

Authors: Imran Abbas Baloch; İmdat İşcan
Journal: International Journal of Analysis
 (2015)

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## Abstract

The authors introduce the concept of harmonicallys,m-convex functions in second sense and establish some Ostrowski type inequalities of these classes of functions.


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## Body

## 1. Introduction

Letf:I⊆R→R be a mapping differentiable in I∘ and a,b∈I with a<b. If |f′(x)|≤M, for all x∈[a,b], then the following inequality holds:(1)fx-1b-a∫abftdt≤Mb-a14+x-a+b/22b-a2for x∈[a,b]. This inequality is known in the literature as the Ostrowski inequality [1], which gives an upper bound for the approximation of the integral average 1/b-a∫ab‍f(t)dt by the value f(x) at the point x∈[a,b]. For some results which generalize, improve, and extend inequality (1), we refer the reader to recent papers (see [2, 3]) and monograph of [4].In [5], İşcan introduced class of harmonically convex functions. This class of functions is defined as follows.Definition 1.
LetI⊂R/{0} be a real interval. A function f:I→R is said to be harmonically convex, if(2)fxytx+1-ty≤tfy+1-tfxfor all x,y∈I and t∈[0,1]. If inequality in (2) is reversed, then f is said to be harmonically concave.In [6], İşcan introduced the concept of harmonically s-convex function in second sense and established a variant of Ostrowski type inequalities which hold for these classes of functions as follows.Definition 2.
A functionf:I⊂R/{0}→R is said to be harmonically s-convex in second sense, if(3)fxytx+1-ty≤tsfy+1-tsfxfor all x,y∈I and t∈[0,1] for some fixed s∈(0,1]. If inequality in (3) is reversed, then f is said to be harmonically s-concave.Remark 3.
Note that, fors=1, harmonically s-convexity reduces to ordinary harmonic convexity.Theorem 4.
Letf:I⊂(0,∞)→R be a differentiable function on I∘, a,b∈I∘ with a<b and f′∈L[a,b]. If |f′|q is harmonically s-convex on [a,b] for q≥1, then, for all x∈[a,b], we have(4)fx-abb-a∫abfuu2du≤abb-ax-a2λ1a,x,s,q,qf′xq+λ2a,x,s,q,qf′aq1/q+b-x2λ3b,x,s,q,qf′xq+λ4b,x,s,q,qf′bq1/q,where(5)λ1a,x,s,ϑ,ρ=βρ+s+1,1x2ϑ·F212ϑ,ρ+s+1;ρ+s+2,1-ax,λ2a,x,s,ϑ,ρ=βρ+1,1x2ϑ·F212ϑ,ρ+1;ρ+s+2,1-ax,λ3b,x,s,ϑ,ρ=β1,ρ+s+1b2ϑ·F212ϑ,1;ρ+s+2,1-xb,λ4b,x,s,ϑ,ρ=βs+1,ρ+1b2ϑ·F212ϑ,s+1;ρ+s+2,1-xb.β is Euler Beta function defined by(6)βx,y=ΓxΓyΓx+y=∫01tx-11-ty-1dt,x,y>0,and F21 is hypergeometric function (see [4]) defined by(7)F21a,b;c,z=1βb,c-b∫01tb-11-tc-b-11-zt-adt,c>b>0,z<1.Corollary 5.
In Theorem4, additionally, if |f′(x)|≤M, x∈[a,b], then inequality(8)fx-abb-a∫abfuu2du≤abb-aMx-a2λ1a,x,s,q,q+λ2a,x,s,q,q1/q+b-x2λ3b,x,s,q,q+λ4b,x,s,q,q1/qholds.Theorem 6.
Letf:I⊂(0,∞)→R be a differentiable function on I∘, a,b∈I∘ with a<b and f′∈L[a,b]. If |f′|q is harmonically s-convex on [a,b] for q≥1, then, for all x∈[a,b], one has(9)fx-abb-a∫abfuu2du≤abb-a121-1/qx-a2λ1a,x,s,q,1f′xq+λ2a,x,s,q,1f′aq1/q+b-x2λ3b,x,s,q,1f′xq+λ4b,x,s,q,1f′bq1/q,where λ1, λ2, λ3, and λ4 are defined as in Theorem 4.Corollary 7.
In Theorem6, additionally, if |f′(x)|≤M, x∈[a,b], then the inequality(10)fx-abb-a∫abfuu2du≤abb-aM121-1/qx-a2λ1a,x,s,q,1+λ2a,x,s,q,11/q+b-x2λ3b,x,s,q,1+λ4b,x,s,q,11/qholds.Theorem 8.
Letf:I⊂(0,∞)→R be a differentiable function on I∘, a,b∈I∘ with a<b and f′∈L[a,b]. If |f′|q is harmonically s-convex on [a,b] for q≥1, then, for all x∈[a,b], one has(11)fx-abb-a∫abfuu2du≤abb-aλ51-1/qa,xx-a2λ1a,x,s,1,1f′xq+λ2a,x,s,1,1f′aq1/q+λ51-1/qb,xb-x2λ3b,x,s,1,1f′xq+λ4b,x,s,1,1f′bq1/q,where(12)λ5θ,x=1x-θ1θ-ln⁡x-ln⁡θx-θand λ1, λ2, λ3, and λ4 are defined as in Theorem 4.Corollary 9.
In Theorem8, additionally, if |f′(x)|≤M, x∈[a,b], then the inequality(13)fx-abb-a∫abfuu2du≤abb-aMλ51-1/qa,xx-a2λ1a,x,s,1,1+λ2a,x,s,1,11/q+λ51-1/qb,xb-x2λ3b,x,s,1,1+λ4b,x,s,1,11/qholds.Theorem 10.
Letf:I⊂(0,∞)→R be a differentiable function on I∘, a,b∈I∘ with a<b and f′∈L[a,b]. If|f′|q is harmonically s-convex on [a,b] for q>1, 1/p+1/q=1, then, for all x∈[a,b], one has(14)fx-abb-a∫abfuu2du≤abb-a11+p1/px-a2λ1a,x,s,q,0f′xq+λ2a,x,s,q,0f′aq1/q+b-x2λ3b,x,s,q,0f′xq+λ4b,x,s,q,0f′bq1/q,where λ1, λ2, λ3, and λ4 are defined as in Theorem 4.Corollary 11.
In Theorem10, additionally, if|f′(x)|≤M, x∈[a,b], then the inequality(15)fx-abb-a∫abfuu2du≤abb-aM11+p1/px-a2λ1a,x,s,q,0+λ2a,x,s,q,01/q+b-x2λ3b,x,s,q,0+λ4b,x,s,q,01/qholds.Theorem 12.
Letf:I⊂(0,∞)→R be a differentiable function on I∘, a,b∈I∘ with a<b and f′∈L[a,b]. If |f′|q is harmonically s-convex on [a,b] for q>1, 1/p+1/q=1, then, for all x∈[a,b], one has(16)fx-abb-a∫abfuu2du≤abb-aλ1a,x,0,p,p1/px-a2f′xq+f′aqs+11/q+λ3b,x,0,p,p1/pb-x2f′xq+f′bqs+11/q,where λ1 and λ3 are defined as in Theorem 4.Corollary 13.
In Theorem12, additionally, if|f′(x)|≤M, x∈[a,b], then the inequality(17)fx-abb-a∫abfuu2du≤abb-aM2s+11/qλ1a,x,0,p,p1/px-a2+λ3b,x,0,p,p1/pb-x2holds.In [7], İşcan introduced the concept of harmonically (α,m)-convex functions and established some Hermite-Hadamard type inequalities for this class of function. This class of functions is defined as follows.Definition 14.
The functionf:(0,∞)→R is said to be harmonically (α,m)-convex, where α∈[0,1] and m∈(0,1], if(18)fmxymty+1-tx=ftx+1-tmy-1≤tαfx+m1-tαfyfor all x,y∈(0,∞) and t∈[0,1]. If the inequality in (18) is reversed, then f is said to be harmonically (α,m)-concave.In [8], Park considered the class of (s,m)-convex functions in second sense. This class of functions is defined as follows.Definition 15.
For some fixeds∈(0,1] and m∈[0,1], a mapping f:I⊂[0,∞)→R is said to be (s,m)-convex in the second sense on I if(19)ftx+m1-ty≤tsfx+m1-tsfyholds, for all x,y∈I and t∈[0,1].Now, we introduce the concept of harmonically(s,m)-convex functions in second sense, which generalize the notion of harmonically convex and harmonically s-convex functions in second sense introduced by İşcan in [5, 6], as follows.Definition 16.
The functionf:I⊂(0,∞)→R is said to be harmonically (s,m)-convex in second sense, where s∈(0,1] and m∈(0,1] if(20)fmxymty+1-tx=ftx+1-tmy-1≤tsfx+m1-tsfy,∀x,y∈I and t∈[0,1].Remark 17.
Note that, fors=1, harmonic (s,m)-convexity reduces to harmonic m-convexity and for m=1 harmonic (s,m)-convexity reduces to harmonic s-convexity in second sense (see [6]) and for s,m=1 harmonic (s,m)-convexity reduces to ordinary harmonic convexity (see [5]).Proposition 18.
Letf:(0,∞)→R be a function:(a)
iff is (s,m)-convex function in second sense and nondecreasing, then f is a harmonically (s,m)-convex function in second sense.(b)
Iff is a harmonically (s,m)-convex function in second sense and nonincreasing, then f is (s,m)-convex function in second sense.Remark 19.
According to Proposition18, every nondecreasing (s,m)-convex function in second sense is also a harmonically (s,m)-convex function in second sense.Example 20 (see [9]).
Let0<s<1 and a,b,c∈R; then, function f:(0,∞)→R defined by(21)fx=a,x=0,bx2+c,x>0,is a nondecreasing s-convex function in second sense for b≥0 and 0≤c≤a. Hence, by Proposition 18, f is a harmonically (s,1)-convex function.Proposition 21.
Lets∈[0,1], m∈(0,1], and f:[a,mb]⊂(0,∞)→R be an increasing function and g:[a,mb]→[a,mb] and g(x)=mab/a+mb-x, a<mb. Then, f is harmonically (s,m)-convex in second sense on [a,mb] if and only if fog is (s,m)-convex in second sense on [a,mb].The following result of the Hermite-Hadamard type holds.Theorem 22.
Letf:I⊂(0,∞)→R be a harmonically (s,m)-convex function in second sense with s∈[0,1] and m∈(0,1]. If0<a<b<∞ and f∈L[a,b], then one has the following inequality:(22)abb-a∫abfxx2dx≤min⁡fa+mfb/ms+1,fb+mfa/ms+1.Corollary 23.
If one takesm=1 in Theorem 22, then one gets(23)abb-a∫abfxx2dx≤fa+fbs+1.Corollary 24.
If one takess=1 in Theorem 22, then one gets(24)abb-a∫abfxx2dx≤min⁡fa+mfb/m2,fb+mfa/m2.In this paper, we obtain similar inequalities (1) for harmonically (s,m)-convex functions and establish some new results of Ostrowski type inequalities for harmonically (s,m)-convex functions such that results given in [6] by İşcan are obtained for particular value of m.


## 2. Main Results

For finding some new inequalities of Ostrowski type for the functions whose derivatives are harmonically(s,m)-convex in second sense, we need the following lemma.Lemma 25.
Letf:I⊂R/{0}→R be a differentiable function on I∘ and a,b∈I with a<b. If f′∈L[a,b], then (25)fx-abb-a∫abfuu2du=abb-ax-a2∫01tta+1-tx2f′axta+1-txdt-b-x2∫01ttb+1-tx2f′bxtb+1-txdt.Theorem 26.
Letf:I⊂(0,∞)→R be a differentiable function on I∘, ma,b∈I∘ with a<b, m∈(0,1], and f′∈L[a,b]. If |f′|q is harmonically (s,m)-convex in second sense on [a,b/m] for q≥1 with s∈[0,1], then(26)fx-abb-a∫abfuu2du≤abb-ax-a2λ1a,x,s,q,qf′xq+mλ2a,x,s,q,qf′amq1/q+b-x2λ3b,x,s,q,qf′xq+mλ4b,x,s,q,qf′bmq1/q,where λ1, λ2, λ3, λ4 are defined as in Theorem 4.Proof.
From Lemma25 and using power mean inequality, we have (27)fx-abb-a∫abfuu2du≤abb-ax-a2∫01tta+1-tx2f′axta+1-txdt+b-x2∫01ttb+1-tx2f′bxtb+1-txdt≤abx-a2b-a∫011dt1-1/q∫01tqta+1-tx2qf′axta+1-txqdt1/q+abb-x2b-a∫011dt1-1/q∫01tqtb+1-tx2qf′bxtb+1-txqdt1/q.Since |f′|q is harmonically (s,m)-convex function in second sense, we have (28)fx-abb-a∫abfuu2du≤abx-a2b-a∫01tqtsf′xq+m1-tsf′a/mqta+1-tx2qdt1/q+abb-x2b-a∫01tqtsf′xq+m1-tsf′b/mqtb+1-tx2qdt1/q.It is easy to check that(29)∫01tq+sta+1-tx2qdt=βq+s+1,1x2q·F212q,q+s+1,q+s+2;1-ax,(30)∫01tq+stb+1-tx2qdt=β1,q+s+1b2q·F212q,1,q+s+2;1-xb,(31)∫01tq1-tsta+1-tx2qdt=βq+1,s+1x2q·F212q,q+1,q+s+2;1-ax,(32)∫01tq1-tstb+1-tx2qdt=βs+1,q+1b2q·F212q,s+1,q+s+2;1-ax.This completes the proof.Remark 27.
If we takem=1 in Theorem 26, we get Theorem 4.Corollary 28.
In Theorem26, additionally, if |f′(x)|≤M, x∈[a,b/m], then inequality (33)fx-abb-a∫abfuu2du≤abb-aMx-a2λ1a,x,s,q,q+mλ2a,x,s,q,q1/q+b-x2λ3b,x,s,q,q+mλ4b,x,s,q,q1/qholds.Remark 29.
If we takem=1 in Corollary 28, we get Corollary 5.Theorem 30.
Letf:I⊂(0,∞)→R be a differentiable function on I, ma,b∈I∘ with a<b, m∈(0,1], and f′∈L[a,b]. If |f′|q is harmonically (s,m)-convex in second sense on [a,b/m] for q≥1 with s∈[0,1], then, for all x∈[a,b], one has(34)fx-abb-a∫abfuu2du≤abb-a121-1/qx-a2λ1a,x,s,q,1f′xq+mλ2a,x,s,q,1f′amq1/q+b-x2λ3b,x,s,q,1f′xq+mλ4b,x,s,q,1f′bmq1/q,where, λ1, λ2, λ3, and λ4 are defined as in Theorem 4.Proof.
From Lemma25, power mean inequality, and harmonic (s,m)-convexity in second sense of |f′|q on [a,b/m], we have (35)fx-abb-a∫abfuu2du≤abb-ax-a2∫01tta+1-tx2f′axta+1-txdt+b-x2∫01ttb+1-tx2f′bxtb+1-txdt≤abx-a2b-a∫01tdt1-1/q∫01tta+1-tx2qf′axta+1-txqdt1/q+abb-x2b-a∫01tdt1-1/q∫01ttb+1-tx2qf′bxtb+1-txqdt1/q≤abx-a2b-a∫01ttsf′xq+m1-tsf′a/mqta+1-tx2qdt1/q+abb-x2b-a∫01ttsf′xq+m1-tsf′b/mqtb+1-tx2qdt1/q≤abb-a121-1/qx-a2λ1a,x,s,q,1f′xq+mλ2a,x,s,q,1f′amq1/q+b-x2λ3b,x,s,q,1f′xq+mλ4b,x,s,q,1f′bmq1/q.
This completes the proof.Remark 31.
If we takem=1 in Theorem 30, we get Theorem 6.Corollary 32.
In Theorem30, additionally, if |f′(x)|≤M, x∈[a,b/m], then inequality (36)fx-abb-a∫abfuu2du≤abb-aM121-1/qx-a2λ1a,x,s,q,1+mλ2a,x,s,q,11/q+b-x2λ3b,x,s,q,1+mλ4b,x,s,q,11/qholds.Remark 33.
If we takem=1 in Corollary 32, then we get Corollary 7.Theorem 34.
Letf:I⊂(0,∞)→R be a differentiable function on I, ma,b∈I∘ with a<b, m∈(0,1], and f′∈L[a,b]. If |f′|q is harmonically (s,m)-convex in second sense on [a,b/m] for q≥1 with s∈[0,1], then, for all x∈[a,b], one has(37)fx-abb-a∫abfuu2du≤abb-aλ51-1/qa,xx-a2λ1a,x,s,1,1f′xq+mλ2a,x,s,1,1f′amq1/q+λ51-1/qb,xb-x2λ3b,x,s,1,1f′xq+mλ4b,x,s,1,1f′bmq1/q,where(38)λ5θ,x=1θ-x1x-θ-ln⁡x-ln⁡ax-θ,and λ1, λ2, λ3, and λ4 are defined as in Theorem 4.Proof.
From Lemma25, power mean inequality, and harmonic (s,m)-convexity in second sense of |f′|q on [a,b/m], we have (39)fx-abb-a∫abfuu2du≤abb-ax-a2∫01tta+1-tx2f′axta+1-txdt+b-x2∫01ttb+1-tx2f′bxtb+1-txdt≤abx-a2b-a∫01tta+1-tx2dt1-1/q∫01tta+1-tx2f′axta+1-txqdt1/q+abb-x2b-a∫01ttb+1-tx2dt1-1/q∫01ttb+1-tx2f′bxtb+1-txqdt1/q≤abx-a2b-a∫01tta+1-tx2dt1-1/q∫01ttsf′xq+m1-tsf′a/mqta+1-tx2qdt1/q+abb-x2b-a∫01ttb+1-tx2dt1-1/q∫01ttsf′xq+m1-tsf′b/mqtb+1-tx2qdt1/q.It is easy to check that(40)∫01tta+1-tx2dt=1x-a1a-ln⁡x-ln⁡ax-a∫01ttb+1-tx2dt=1b-xln⁡x-ln⁡ax-a-1b.Hence, by use of (29), (30), (31), and (32) for q=1 and (40) in (39), we get the desired result.Remark 35.
If we takem=1 in Theorem 34, we get Theorem 8.Corollary 36.
In Theorem34, additionally, if |f′(x)|≤M, x∈[a,b/m], then inequality (41)fx-abb-a∫abfuu2du≤abb-aMλ51-1/qa,xx-a2λ1a,x,s,1,1+mλ2a,x,s,1,11/q+λ51-1/qb,xb-x2λ3b,x,s,1,1+mλ4b,x,s,1,11/qholds.Remark 37.
If we takem=1 in Corollary 36, we get Corollary 9.Theorem 38.
Letf:I⊂(0,∞)→R be a differentiable function on I∘, ma,b∈I∘ with a<b, m∈(0,1], and f′∈L[a,b]. If |f′|q is harmonically (s,m)-convex in second sense on [a,b/m] for q>1, 1/p+1/q=1, with s∈[0,1], then (42)fx-abb-a∫abfuu2du≤abb-a1p+11/px-a2λ1a,x,s,q,0f′xq+mλ2a,x,s,q,0f′amq1/q+b-x2λ3b,x,s,q,0f′xq+mλ4b,x,s,q,0f′bmq1/q,where, λ1, λ2, λ3, and λ4 are defined as in Theorem 4.Proof.
From Lemma25, Hölder’s inequality, and harmonic (s,m)-convexity of |f′|q on [a,b/m], one has (43)fx-abb-a∫abfuu2du≤abx-a2b-a∫01tpdt1/p∫011ta+1-tx2qf′axta+1-txqdt1/q+abb-x2b-a∫01tpdt1/p∫011tb+1-tx2qf′bxtb+1-txqdt1/q≤abx-a2b-a1p+11/p∫01tsf′xq+m1-tsf′a/mqta+1-tx2qdt1/q+abb-x2b-a∫01tsf′xq+m1-tsf′b/mqtb+1-tx2qdt1/q≤abb-a1p+11/px-a2λ1a,x,s,q,0f′xq+mλ2a,x,s,q,0f′amq1/q+b-x2λ3b,x,s,q,0f′xq+mλ4b,x,s,q,0f′bmq1/q.This completes the proof.Remark 39.
If we takem=1 in Theorem 38, we get Theorem 10.Corollary 40.
In Theorem38, additionally, if |f′(x)|≤M, x∈[a,b/m], then inequality (44)fx-abb-a∫abfuu2dx≤abb-aM1p+11/px-a2λ1a,x,s,q,0+mλ2a,x,s,q,01/q+b-x2λ3b,x,s,q,0+mλ4b,x,s,q,01/qholds.Remark 41.
If we takem=1 in Corollary 40, we get Corollary 11.Theorem 42.
Letf:I⊂(0,∞)→R be a differentiable function on I∘, ma,b∈I∘ with a<b, m∈(0,1], and f′∈L[a,b]. If |f′|q is harmonically (s,m)-convex in second sense on [a,b/m] for q>1, 1/p+1/q=1, with s∈[0,1], then one has the following inequality:(45)fx-abb-a∫abfuu2du≤abb-aλ1a,x,0,p,px-a2f′xq+mf′a/mqs+11/q+λ3b,x,0,p,pb-x2f′xq+mf′b/mqs+11/q,where, λ1 and λ3 are defined as in Theorem 4.Proof.
From Lemma25, Hölder’s inequality, and harmonic (s,m)-convexity of |f′|q on [a,b/m], we have (46)fx-abb-a∫abfuu2du≤abx-a2b-a∫01tpta+1-tx2pdt1/p∫01tsf′xq+m1-tsf′amq1/q+abb-x2b-a∫01tptb+1-tx2pdt1/p∫01tsf′xq+m1-tsf′bmq1/q≤abb-aλ1a,x,0,p,px-a2f′xq+mf′a/mqs+11/q+λ3b,x,0,p,pb-x2f′xq+mf′b/mqs+11/q.Remark 43.
If we takem=1 in Theorem 42, we get Theorem 12.Corollary 44.
In Theorem42, additionally, if |f′(x)|≤M, x∈[a,b/m], then inequality (47)fx-abb-a∫abfuu2du≤abb-aM2s+11/qλ1a,x,0,p,p1/px-a2+λ3b,x,0,p,p1/pb-x2holds.Remark 45.
Corollary44 is exactly as Corollary 13.



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