# Some Integral Inequalities for Harmonically(α,s)-Convex Functions

Authors: Serap Özcan
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In the paper, the author introduces a new class of harmonically convex functions, which is called harmonicallyα,s-convex functions and establishes some new integral inequalities of the Hermite-Hadamard type for harmonically α,s-convex functions. The properties of the newly introduced class of harmonically convex functions are also investigated. Finally, some applications to special means are given.


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## Body

## 1. Introduction

It is well known in the literature that a functionf:I⊆R→R is said to be convex on interval I if(1)ftx+1-ty≤tfx+1-tfyfor all x,y∈I and t∈0,1. If the inequality in (1) holds in the reverse direction, f is said to be concave function.Inequalities play a fundamental part in many branches of pure and applied sciences. A number of studies have shown that the theory of convex functions has a closely relationship with the theory of inequalities. One of the most famous inequalities for convex functions is named Hermite-Hadamard integral inequality as follows.Supposef:I⊆R→R to be a convex function defined on the interval I of real numbers and a,b∈I with a<b; then the double inequality(2)fa+b2≤1b-a∫abfxdx≤fa+fb2holds. If f is concave function, both inequalities in (2) are reversed. This inequality provides necessary and sufficient condition for a function to be convex.In recent years, the concept of convexity has been improved, generalized, and extended in many directions; see [1–9]. Several new classes of convex functions have been introduced and new versions of Hermite-Hadamard’s inequality have been obtained. In [10], I˙șcan introduced the class of harmonically convex functions and investigated the Hermite-Hadamard type inequalities for this new class of functions. For several recent results, generalizations, improvements, and refinements concerning harmonically convex functions; see [10–16]. There have been many studies dedicated to generalizing the harmonic convex functions and to establishing their Hermite-Hadamard type inequalities. For some recent studies on Hermite-Hadamard type inequalities, please refer to the monographs [10–12, 16–24].The aim of this paper is to introduce the concept of harmonicallyα,s-convex functions and to establish several new Hermite-Hadamard type inequalities based on these new class of functions.


## 2. Preliminaries

In this section, we recall some basic concepts and results of harmonically convex functions.Definition 1 ([10]).
LetI⊂0,∞ be a real interval. A function f:I→R is said to be harmonically convex function if(3)fxytx+1-ty=tfy+1-tfx for all x,y∈I and t∈0,1.Definition 2 ([13]).
A functionf:I⊂0,∞→R is said to be harmonically s-convex function in second sense, where s∈0,1, if(4)fxytx+1-ty=tsfy+1-tsfx for all x,y∈I and t∈0,1.Definition 3 ([15]).
A functionf:I⊂0,∞→R is said to be harmonically s-convex function in first sense, where s∈0,1, if(5)fxytx+1-ty=tsfy+1-tsfx for all x,y∈I and t∈0,1.Definition 4 ([12]).
The functionf:0,b∗→R, b∗>0, is said to be harmonically α,m-convex function, where α∈0,1 and m∈0,1, if(6)fmxymtx+1-ty=fty+1-tmx-1≤tαfy+m1-tαfx for all x,y∈0,b∗ and t∈0,1.Note that ifm=1 in Definition 4, we have definition of harmonically s-convex function in first sense for α=s.Definition 5 ([25]).
A functionf:I⊂0,∞→R is said to be harmonically P-function if(7)fxytx+1-ty=fx+fy for all x,y∈I and t∈0,1.Theorem 6 ([10]).
LetI⊂0,∞→R be harmonically convex and a,b∈I with a<b. If f∈La,b, then the following inequalities hold:(8)f2aba+b≤abb-a∫abfxx2dx≤fa+fb2.Theorem 7 ([10]).
Letf:I⊂0,∞→R be a differentiable function on I∘, a,b∈I with a<b and f′∈La,b. If f′q is harmonically convex on a,b for q≥1, then(9)fa+fb2-abb-a∫abfxx2dx≤abb-a2λ11-1/qλ2f′aq+λ3f′bq1/q,where(10)λ1=1ab-2b-a2ln⁡a+b24ab,(11)λ2=-1bb-a+3a+bb-a3ln⁡a+b24ab,and (12)λ3=1ab-a+3b+ab-a3ln⁡a+b24ab.Theorem 8 ([10]).
Letf:I⊂0,∞→R be a differentiable function on I∘, a,b∈I with a<b and f′∈La,b. If f′q is harmonically convex on a,b for p,q>1, 1/p+1/q=1, then(13)fa+fb2-abb-a∫abfxx2dx≤abb-a21p+11/pμ1f′aq+μ2f′bq1/q,where(14)μ1=a2-2q+b1-2qb-a1-2q-a2b-a21-q1-2q,μ2=b2-2q-a1-2qb-a1-2q+b2b-a21-q1-2q.Definition 9 ([9]).
Letf and g be two functions. If(15)fx-fygx-gy≥0 holds for all x,y∈R, then f and g are said to be similarly ordered functions.Definition 10 ([23]).
A functionf:I⊂R→R is said to be α,s-convex if(16)ftx+1-ty≤tαsfx+1-tαsfy for all x,y∈I and t∈0,1 with s∈-1,1 and α∈0,1.In order to prove some of our main results, we need the following lemma.Lemma 11 ([10]).
Letf:I⊂0,∞→R be a differentiable function on I∘ and a,b∈I with a<b. If f′∈La,b, then(17)fa+fb2-abb-a∫abfxx2dx=abb-a2∫011-2ttb+1-ta2f′abtb+1-tadt


## 3. Main Results

In this section we define the concept of harmonicallyα,s-convex functions and derive our main results. Throughout this section I⊂0,∞ is the interval and I∘ is the interior of I.Definition 12.
A functionf:I→R is said to be harmonically α,s-convex, where α,s∈0,12, if(18)fxyty+1-tx=ftx+1-ty-1≤tαsfx+1-tαsfyfor all x,y∈I and t∈0,1. If the inequality in (18) is reversed, then f is said to be harmonically α,s-concave.Now we give some special cases of our proposed definition of harmonicallyα,s-convex functions.(I) Ifα=1, then we have the definition of harmonically s-convex functions in second sense.(II) Ifs=1, then we have the definition of harmonically α,1-convex functions or in other words harmonically s-convex functions in first sense by writing s instead of α.(III) Ifα=s=1, then we have the definition of harmonically convex functions. Thus, every harmonically convex function is also harmonically 1,1-convex function.The following proposition is obvious.Proposition 13.
Letf:I⊂0,∞→R be a function.(1)
Iff is α,s-convex and nondecreasing function, then f is harmonically α,s-convex. (2)
Iff is harmonically α,s-convex and nonincreasing function, then f is α,s-convex.Proof.
For allt∈0,1 and x,y∈I, we have(19)t1-tx-y2≥0. So, the following inequality holds:(20)xyty+1-tx≤tx+1-ty.By inequality (20), the proof is completed.Remark 14.
According to Proposition13, every nondecreasing convex function (or 1,1-convex function) is also harmonically convex function (or harmonically 1,1-convex function).Example 15.
The functionf:0,∞→R, fx=x is a nondecreasing 1,1-convex function. According to Remark 14, f is harmonically 1,1-convex function (or harmonically α,s-convex function for α=s=1).Proposition 16.
Letf and g be two harmonically α,s-convex functions. If f and g are similarly ordered functions and tαs+1-tαs≤1, then the product fg is also harmonically α,s-convex function.Proof.
Letf and g be harmonically α,s-convex functions. Then(21)fabtb+1-tagabtb+1-ta≤tαsfa+1-tαsfbtαsga+1-tαsgb=t2αsfaga+tαs1-tαsfagb+fbga+1-tα2sfbgb≤t2αsfaga+tαs1-tαsfaga+fbgb+1-tα2sfbgb=tαsfaga+1-tαsfbgbtαs+1-tαs≤tαsfaga+1-tαsfbgb.Proposition 17.
Letf:a,b⊂0,∞→R and g:1/b,1/a→R defined by gx=1/x; then f is harmonically α,s-convex on a,b, where α∈0,1, s∈0,1, if and only if g is α,s-convex on 1/b,1/a.Proof.
Since(22)f∘gtx+1-ty=f1tx+1-ty=f1t/r+1-t/k=ftr+1-tk-1≤tαsfr+1-tαsfk=tαsf∘gx+1-tαsf∘gy for all x,y∈1/b,1/a, t∈0,1, where r,k∈a,b, x=1/r, y=1/k.
This demonstrates necessary condition is provided. Now let us show the sufficient condition is also provided.
For allx,y∈a,b and t∈0,1, we have (23)ftx+1-ty-1=f∘gt.1x+1-t.1y≤tαsf∘g1x+1-tαsf∘g1y=tαsfx+1-tαsfy. This completes the proof.Theorem 18.
Letf:I→R be harmonically α,s-convex function with α∈0,1, s∈0,1. If 0<a<b<∞ and f∈La,b, then(24)abb-a∫abfxx2dx≤fa+fb2∫01tαs+1-tαsdt.Proof.
Sincef is a harmonically α,s-convex function, we have for all x,y∈I(25)fxyty+1-tx≤tαsfx+1-tαsfy which gives (26)fabtb+1-ta≤tαsfa+1-tαsfb,fabta+1-tb≤tαsfb+1-tαsfa for all t∈0,1.
Adding the above inequalities, we have(27)fabtb+1-ta+fabta+1-tb≤fa+fbtαs+1-tαs
Integrating the above inequality on0,1, we obtain(28)∫01fabtb+1-tadt+∫01fabta+1-tbdt≤fa+fb∫01tαs+1-tαsdt which implies(29)abb-a∫abfxx2dx≤fa+fb2∫01tαs+1-tαsdt which is the required result.Remark 19.
If we takeα=s=1 in Theorem 18, then inequality (24) becomes the right-hand side of inequality (2).If we takeα=1 in Theorem 18, we have the following result for harmonically s-convex functions in second sense.Corollary 20.
Letf:I→R be harmonically s-convex function in second sense where a,b∈I with a<b and s∈0,1. If f∈La,b, then(30)abb-a∫abfxx2dx≤fa+fbs+1.If we takes=0, then Theorem 18 collapses to the following result.Corollary 21.
Letf:I→R be harmonically P-function where a,b∈I with a<b. If f∈La,b, then(31)abb-a∫abfxx2dx≤fa+fb.Our coming result is the Hermite-Hadamard inequality for product of two harmonicallyα,s-convex functions.Theorem 22.
Letf,g:I→R be two harmonically α,s-convex functions where a,b∈I with a<b, α∈0,1, s∈0,1. If fg∈La,b, then(32)abb-a∫abfxgxx2dx≤η1M1+η2M2+η3M3, where(33)η1=faga,(34)η2=fbgb,(35)η3=fagb+fbga,and (36)M1=∫01t2αsdt=12αs+1,M2=∫011-tα2sdt=β2s+1,1α+1,M3=∫01tαs1-tαsdt.β is Euler Beta function defined by(37)βx,y=∫01tx-11-ty-1dt,x,y>0, and F12 is hypergeometric function defined by(38)F12a,b,c;z=1βb,c-b∫01tb-11-tc-b-11-zt-adt,c>b>0,z<1.Proof.
Letf,g:I→R be harmonically α,s-convex functions; then we have (39)abb-a∫abfxgxx2dx=∫01fxyty+1-txgxyty+1-txdt≤∫01tαsfa+1-tαsfbtαsga+1-tαsgbdt=∫01t2αsfaga+1-tα2sfbgb+tαs1-tαsfagb+fbgadt=η1∫01t2αsdt+η2∫011-tα2sdt+η3∫01tαs1-tαsdt=η1M1+η2M2+η3M3. This completes the proof.Theorem 23.
Under the conditions of Theorem22, if f and g are similarly ordered functions, then we have(40)abb-a∫abfxgxx2dx≤η1N1+η2N2, where η1 and η2 are given by (33) and (34), (41)N1=∫01tαsdt=1αs+1,N2=∫011-tαsdt=βs+1,1α+1.Proof.
Integrating inequality (21) completes the proof.Theorem 24.
Letf:I→R be a differentiable function on I∘, a,b∈I with a<b and f′∈La,b. If f′q is harmonically α,s-convex on a,b for q≥1, with α∈0,1 and s∈0,1, then(42)fa+fb2-abb-a∫abfxx2dx≤abb-a2λ11-1/qκ1f′aq+κ2f′bq1/q,where λ1 is given by (10),(43)κ1=∫011-2ttαstb+1-ta2dt,and(44)κ2=∫011-2t1-tαstb+1-ta2dt.Proof.
Using Lemma11 and power mean inequality, we have (45)fa+fb2-abb-a∫abfxx2dx≤abb-a2∫011-2ttb+1-ta2f′abtb+1-tadt≤abb-a2∫011-2ttb+1-ta2dt1-1/q×∫011-2ttb+1-ta2f′abtb+1-taqdt1/q≤abb-a2∫011-2ttb+1-ta2dt1-1/q×1-2ttb+1-ta2tαsf′aq+1-tαsf′bqdt1/q=abb-a2λ11-1/qκ1f′aq+κ2f′bq1/q. This completes the proof.Corollary 25.
Under the conditions of Theorem24 if q=1, then we have(46)fa+fb2-abb-a∫abfxx2dx≤abb-a2κ1f′a+κ2f′b, where κ1 and κ2 are given by (43) and (44), respectively.Remark 26.
If we takeα=s=1 in Theorem 24, then inequality (42) becomes inequality (9) of Theorem 7.If we takeα=1 in Theorem 24, then we have result for harmonically s-convex functions in second sense.Corollary 27.
Letf:I→R be a differentiable function on I∘, a,b∈I with a<b and f′∈La,b. If f′q is harmonically s-convex in second sense on a,b for q>1, then(47)fa+fb2-abb-a∫abfxx2dx≤abb-a2λ11-1/qρ1f′aq+ρ2f′bq1/q,where λ1 is given by (10),(48)ρ1=∫011-2ttstb+1-ta2dt,and(49)ρ2=∫011-2t1-tstb+1-ta2dt,respectively.If we takes=1 in Theorem 24, then we have following result for harmonically s-convex functions in first sense for α=s.Corollary 28.
Letf:I→R be a differentiable function on I∘, a,b∈I with a<b and f′∈La,b. If f′q is harmonically s-convex function in first sense, then(50)fa+fb2-abb-a∫abfxx2dx≤abb-a2λ11-1/qρ1f′aq+ρ∗f′bq1/q,where λ1 is given by (10), and(51)ρ1=∫011-2ttstb+1-ta2dt, and(52)ρ∗=λ1-ρ1.If we takes=0 in Theorem 24, we have the following result for harmonically P -functions.Corollary 29.
Letf:I→R be a differentiable function on I∘, a,b∈I with a<b and f′∈La,b. If f′q is harmonically P-function for q≥1, then(53)fa+fb2-abb-a∫abfxx2dx≤abb-a2λ1f′aq+f′bq1/q, where λ1 is given by (10).Theorem 30.
Letf:I→R be a differentiable function on I∘, a,b∈I∘ with a<b and f′∈La,b. If f′q is harmonically α,s-convex on a,b for p,q>1, 1/p+1/q=1, with α∈0,1, s∈0,1, then(54)fa+fb2-abb-a∫abfxx2dx≤abb-a21p+11/pψ1f′aq+ψ2f′bq1/q,where(55)ψ1=∫01tαstb+1-ta2qdt, and(56)ψ2=∫011-tαstb+1-ta2qdt, respectively.Proof.
From Lemma11, Hölder’s inequality, and the fact that f′q is harmonically α,s -convex function on a,b, we have (57)fa+fb2-abb-a∫abfxx2dx≤abb-a2∫011-2ttb+1-ta2f′abtb+1-tadt≤abb-a2∫011-2tpdt1/p∫011tb+1-ta2qf′abtb+1-taqdt1/q≤abb-a21p+11/p∫011tb+1-ta2qtαsf′aq+1-tαsf′bqdt1/q=abb-a21p+11/pψ1f′aq+ψ2f′bq1/q where(58)∫011-2tpdt=1p+1. This completes the proof.Remark 31.
If we takeα=s=1 in Theorem 30, then inequality (54) becomes inequality (13) of Theorem 8.If we takeα=1 in Theorem 30, then we have result for harmonically s-convex functions in second sense.Corollary 32.
Letf:I→R be a differentiable function on I∘, a,b∈I∘ with a<b and f′∈La,b. If f′q is harmonically s-convex function in second sense on a,b for p,q>1, 1/p+1/q=1, with s∈0,1, then(59)fa+fb2-abb-a∫abfxx2dx≤abb-a21p+11/pφ1f′aq+φ2f′bq1/q, where(60)φ1=∫01tstb+1-ta2qdt=β1,s+1b2q.2F12q,1,s+2,1-ab,and(61)φ2=∫011-tstb+1-ta2qdt=βs+1,1b2q.2F12q,s+1,s+2,1-ab,respectively.If we takes=1 in Theorem 30, then we have the following result for harmonically s-convex functions in first sense for α=s.Corollary 33.
Letf:I→R be a differentiable function on I∘, a,b∈I∘ with a<b and f′∈La,b. If f′q is harmonically s-convex in first sense on a,b for p,q>1, 1/p+1/q=1, with s∈0,1, then(62)fa+fb2-abb-a∫abfxx2dx≤abb-a21p+11/pϑ1f′aq+ϑ2f′bq1/q,where(63)ϑ1=∫01tstb+1-ta2qdt=β1,s+1b2q.2F12q,1,s+2,1-ab,and(64)ϑ2=∫011-tstb+1-ta2qdt=b1-2q-a1-2qb-a1-2q-ϑ1,respectively.Ifs=0 in Theorem 30, then we have result for harmonically P-functions.Corollary 34.
Letf:I→R be a differentiable function on I∘, a,b∈I∘ with a<b and f′∈La,b. If f′q is harmonically P-function on a,b for p,q>1, 1/p+1/q=1, then(65)fa+fb2-abb-a∫abfxx2dx≤abb-a21p+11/pυ1/qf′aq+f′bq1/q,where(66)υ=∫011tb+1-ta2qdt=b1-2q-a1-2qb-a1-2q.


## 4. Some Applications to Special Means

Let us recall the following special means for arbitrary real numbersa and ba≠b.( 1 ) The arithmetic mean: (67)A≔Aa,b=a+b2.( 2 ) The geometric mean: (68)G≔Ga,b=ab.( 3 ) The p-logarithmic mean: (69)Lp≔Lpa,b=bp+1-ap+1p+1b-a1/p,p∈R∖-1,0.Proposition 35.
Let0<a<b, q>1 and s∈0,1. Then, we have (70)Aas/q+1,bs/q+1-G2Ls/q-1s/q-1≤abb-as+q2qλ11-1/qρ1as+ρ2bs1/q where λ1, ρ1, and ρ2 are given by (10), (48), and (49), respectively.Proof.
The assertion follows from inequality (47) in Corollary 27, for :0,∞→R, fx=xs/q+1/s/q+1.Proposition 36.
Let0<a<b, p>1, q=p/(p-1), and s∈0,1. Then, we have (71)Aas/q+1,bs/q+1-G2Ls/q-1s/q-1≤abb-as+q2q1p+11/pϑ1as+ϑ2bs1/q where ϑ1 and ϑ2 are given by (63) and (64).Proof.
The assertion follows from inequality (62) in Corollary 33, for :0,∞→R, fx=xs/q+1/s/q+1.


## 5. Conclusion

In this paper, we introduce and investigate a new class of harmonically convex functions, which is called harmonicallyα,s-convex functions. Some new results of Hermite-Hadamard type for the newly introduced class of harmonically convex functions are established. Some applications of these results to special means have also been presented. The results of this paper may stimulate further research for the researchers working in this field.



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