# Some Hermite-Hadamard-Fejér Type Integral Inequalities for Differentiableη-Convex Functions with Applications

Authors: M. Rostamian Delavar; M. De La Sen
Journal: Journal of Mathematics
 (2017)

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## Abstract

Using some results about generalized Hermite-Hadamard-Fejér type inequalities related toη-convex functions, we give some examples and applications for trapezoid and midpoint type inequalities for differentiable η-convex functions.


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## Body

## 1. Introduction and Preliminaries

The celebrated Hermite-Hadamard-Fejér inequality (simply Fejér inequality) for convex functions has been proved in [1] as the following.Theorem 1.
Letf:[a,b]→R be a convex function. Then(1)fa+b2∫abgxdx≤∫abfxgxdx≤fa+fb2∫abgxdx,where g:[a,b]→R+=[0,+∞) is integrable and symmetric about x=a+b/2 (g(x)=g(a+b-x),∀x∈[a,b]).If in (1) we consider g≡1, then we obtain Hermite-Hadamard inequality.(2)fa+b2≤1b-a∫abfxdx≤fa+fb2.For various types of (1) and more results related to generalized Hermite-Hadamard-Fejér inequality, see [2–8] and references therein.On the other hand, the concept ofη-convex functions, firstly named by φ-convex functions, as generalization of convex functions has been introduced in [9].LetI be an interval in real line R. Consider η:A×A→B for appropriate A,B⊆R.Definition 2 (see [9, 10]).
A functionf:I→R is called convex with respect to η (briefly η-convex), if(3)ftx+1-ty≤fy+tηfx,fy,for all x,y∈I and t∈[0,1].In fact, the above definition geometrically says that if a function isη-convex on I, then its graph between any x,y∈I is on or under the path starting from (y,f(y)) and ending at (x,f(y)+η(f(x),f(y))). If f(x) should be the end point of the path for every x,y∈I, then we have η(x,y)=x-y and the function reduces to a convex one. Note that by taking x=y in (1) we get tη(f(x),f(x))≥0 for any x∈I and t∈[0,1] which implies that (4)ηfx,fx≥0 for any x∈I. Also if we take t = 1 in (1) we get (5)fx-fy≤ηfx,fyfor any x,y∈I.There are simple examples aboutη-convexity of a function.Example 3 (see [9, 10]).
(
1
) Consider a function f:R→R defined as (6)fx=-x,x≥0;x,x<0, and define a bifunction η as η(x,y)=-x-y, for all x,y∈R-=(-∞,0]. It is not hard to check that f is a η-convex function but not a convex one.
(
2
) Define the function f:R+→R+ as (7)fx=x,0≤x≤1;1,x>1,and a bifunction η:R+×R+→R+ as (8)ηx,y=x+y,x≤y;2x+y,x>y.
Thenf is η-convex but is not convex.The following characterization ofη-convexity holds [11].Theorem 4.
A functionf:I→R is η-convex if and only if, for any x1,x2,x3∈I with x1<x2<x3,(9)det⁡1x1ηfx1,fx31x2fx2-fx31x30≥0,fx1≤fx3+ηfx1,fx3.The following result is of importance.Theorem 5 (see [9, 11]).
Suppose thatf:I→R is a η-convex function and η is bounded from above on f(I)×f(I). Then f satisfies a Lipschitz condition on any closed interval [a,b] contained in the interior I∘ of I. Hence, f is absolutely continuous on [a,b] and continuous on I∘.Note. As a consequence of Theorem 5, a η-convex function f:[a,b]→R where η is bounded from above on f([a,b])×f([a,b]) is integrable. For more results about η-convex functions, see [9–12].Motivated by the above works, in this paper we give some applications and examples for trapezoidal and midpoint type inequalities when the intended function is differentiable. Furthermore we consider integral quadrature formula and give an error estimate related to trapezoidal and midpoint formula. Furthermore some examples support our results.The following two theorems have been proved in [13] which improve the right part and the lefty side of (1), respectively.Theorem 6 (see [13]).
Letf:[a,b]→R be a η-convex function which η is bounded from above on f([a,b])×f([a,b]). If g:[a,b]→R+ is integrable on [a,b], then we have inequalities(10)12∫abfx+fa+b-xgxdx≤min⁡fb+12ηfa,fb,fa+12ηfb,fa∫abgxdx≤12fa+fb∫abgxdx+14ηfa,fb+ηfb,fa∫abgxdx.
Ifg is symmetric on a,b, then from inequality (10) we get(11)∫abfxgxdx≤min⁡fb+12ηfa,fb,fa+12ηfb,fa∫abgxdx≤12fa+fb∫abgxdx+14ηfa,fb+ηfb,fa∫abgxdx.Theorem 7 (see [13]).
Letf:[a,b]→R be a η-convex function with η bounded from above on f([a,b])×f([a,b]). If g:[a,b]→R+ is integrable on [a,b], then(12)fa+b2∫abgxdx≤∫abgxmin⁡fa+b-x+12ηfx,fa+b-x,fx+12ηfa+b-x,fxdx≤min⁡∫abgxfa+b-xdx+12∫abgxηfx,fa+b-xdx,∫abgxfxdx+12∫abgxηfa+b-x,fxdx.Moreover, if g is symmetric on a,b, then(13)fa+b2∫abgxdx≤∫abgxmin⁡fa+b-x+12ηfx,fa+b-x,fx+12ηfa+b-x,fxdx≤∫abgxfxdx+12∫abgxηfx,fa+b-xdx.Remark 8.
(a) Inequality (11) gives a refinement for the right side and inequality (13) gives a refinement for the left side of (1), respectively.
(b) If in Theorems6 and 7 we consider g(x)≡1 for all x∈[a,b] (see Theorem  3.6 in [9]), then respectively, we have (14)1b-a∫abfxdx≤min⁡fa+12ηfb,fa,fb+12ηfa,fb≤fa+fb2+ηfa,fb+ηfb,fa4, which is a refinement for the right side of Hermite-Hadamard inequality related to η-convex functions, and (15)fa+b2-12b-a∫abηfx,fa+b-xdx≤b-amin⁡fa+b-x+12ηfx,fa+b-x,fx+12ηfa+b-x,fxdx≤1b-a∫abfxdx, which is a refinement for the left side of Hermite-Hadamard inequality related to η-convex functions.


## 2. Main Results

As an application of Theorem6, we can obtain some estimation results for the difference between the right and middle part of (2) and also for the difference between the left and middle part of (2), respectively.The following identity for an absolutely continuous functionf:a,b→R holds (see [14]):(16)fa+fb2b-a-∫abftdt=∫abt-a+b2f′tdt.Theorem 9.
Letf:I∘⊂I→R be a differentiable function, a,b∈I∘,a<b. Suppose that the function f′ is a η-convex function and η is bounded from above on f′([a,b])×f′([a,b]), and then we have(17)fa+fb2b-a-∫abftdt≤14b-a2×min⁡f′b+12ηf′a,f′b,f′a+12ηf′b,f′a≤18b-a2×f′a+f′b+12ηf′a,f′b+ηf′b,f′a.Proof.
From (16) we get(18)fa+fb2b-a-∫abftdt≤∫abt-a+b2f′tdt.Since f′ is a η-convex function and g:a,b→R with gt≔t-a+b/2 is symmetric, then by inequality (11) we have(19)∫abf′tt-a+b2dt≤min⁡f′b+12ηf′a,f′b,f′a+12ηf′b,f′a×∫abt-a+b2dt≤12f′a+f′b∫abt-a+b2dt+14ηf′a,f′b+ηf′b,f′a∫abt-a+b2dt.A simple calculation shows that (20)∫abt-a+b2dt=14b-a2;then by (19) we get the desired result (17).Also there exists another identity for absolutely continuous functionsf:a,b→R as the following (see [15]).(21)b-afa+b2-∫abftdt=∫abHtf′tdt,where H:a,b→R is considered as (22)Ht≔t-a,t∈a,a+b2,t-b,t∈a+b2,b. So we can obtain a result for the difference between the left and middle part of (2).Theorem 10.
Letf:I∘⊂I→R be a differentiable function, a,b∈I∘,a<b. Suppose that the function f′ is a η-convex function and η is bounded from above on f′([a,b])×f′([a,b]), and then (23)b-afa+b2-∫abftdt≤14b-a2×min⁡f′b+12ηf′a,f′b,f′a+12ηf′b,f′a≤18b-a2×f′a+f′b+12ηf′a,f′b+ηf′b,f′a.Proof.
From (21) we have (24)b-afa+b2-∫abftdt≤∫abHtf′tdt.It follows that(25)Ht=t-a,t∈a,a+b2,b-t,t∈a+b2,b,which is a symmetric function on the interval a,b. Since f′ is a η-convex function and g:a,b→R with gt≔Ht is symmetric, then by inequality (11) we have(26)∫abf′tHtdt≤min⁡f′b+12ηf′a,f′b,f′a+12ηf′b,f′a×∫abHtdt≤12f′a+f′b∫abHtdt+14ηf′a,f′b+ηf′b,f′a∫abHtdt.Since(27)∫abHtdt=14b-a2,then by (26) we get the desired result (23).Example 11.
Consider the functionF:R+→R+ defined as (28)Fx=12x2+12,0≤x≤1;x,x>1, and a bifunction η:R+×R+→R+ as (29)ηx,y=x+y,x≤y;2x+y,x>y. It is clear that the function (30)fx=F′x=x,0≤x≤1;1,x>1,is a η-convex function. From inequality (23) in Theorem 10, for 0≤a<1<b≤2 with a+b=2, we get(31)∫abFxdx-F1b-a≤14b-a2min⁡1+12ηa,1,a+12η1,a.
Further calculations show that(32)a3-3b2-3a+6b-1≤32Kb-a2, where (33)K=2a+1,a≥13;a+32,a<13.
Also with the same argument as above, from (17) in Theorem 9, for 0≤a<1<b≤2 with a+b=2, we can obtain the following inequality: (34)Fa+Fb2b-a-∫abFxdx≤14b-a2min⁡1+12ηa,1,a+12η1,a,which implies that (35)-5a3+3a2b-6ab-9a+12b2+3b+2≤3Kb-a2, where K is defined as above.Using the following identities for twice differentiable functionsf:a,b→R, we can obtain some results similar to Theorems 9 and 10.(36)fa+fb2b-a-∫abftdt=12∫abt-ab-tf′′tdt;see [14];(37)∫abftdt-b-afa+b2=12∫abGtf′′tdt;see [16], where(38)Gt≔t-a2,t∈a,a+b2,t-b2,t∈a+b2,b; then we have the following.Theorem 12.
Let the functionf:I∘⊂I→R be twice differentiable, a,b∈I∘,a<b. Suppose that the function f′′ is a η-convex function and η is bounded from above on f′′([a,b])×f′′([a,b]), and then we have the inequalities(39)fa+fb2-1b-a∫abftdt≤112b-a2×min⁡f′′b+12ηf′′a,f′′b,f′′a+12ηf′′b,f′′a≤124b-a2×f′′a+f′′b+12ηf′′a,f′′b+ηf′′b,f′′a.Proof.
Taking the modulus on (36) along with (40)∫abt-ab-tdt=16b-a3 implies the expected result (39).Theorem 13.
With the assumptions of Theorem12, we have the inequalities(41)fa+b2-1b-a∫abftdt≤124b-a2×min⁡f′′b+12ηf′′a,f′′b,f′′a+12ηf′′b,f′′a≤148b-a2×f′′a+f′′b+12ηf′′a,f′′b+ηf′′b,f′′a.Proof.
Taking the modulus on (37) along with (42)∫abGtdt=112b-a3, implies the expected result (41).Example 14.
Consider the functionF:R+→R+ defined as (43)Fx=16x3+12x-16,0≤x≤1;12x2,x>1 and the bifunction η defined in Example 11. It is easy to see that the function f(x)=F′′ is a η-convex function. From inequality (41) in Theorem 13, for 0≤a<1<b≤2 with a+b=2, we get(44)∫abFxdx-F1b-a≤124b-a3min⁡1+12ηa,1,a+12η1,a.
It follows that(45)a4-4b3+6a2+12b-16a+1≤Kb-a3, where K is defined in Example 11.In [13], we estimated the difference of the left and middle section of (1) as the following.Theorem 15 (see [13]).
Suppose thatf:I∘⊂I→R is a differentiable mapping, a,b∈I∘,a<b. If g:[a,b]→R+ is a continuous mapping symmetric about a+b/2 and f′ is a η-convex mapping on [a,b] with η bounded from above on f([a,b])×f([a,b]), then(46)∫abfxgxdx-fa+b2∫abgxdx≤1b-a∫a+b/2bx-a2-b-x2gxKdx,where (47)K=min⁡f′b+ηf′a,f′b2,f′a+ηf′b,f′a2.Theorem 16 (see [10]).
Suppose thatf:I∘⊂I→R is a differentiable mapping, a,b∈I∘,a<b. If g:[a,b]→R+ is a continuous function and symmetric about a+b/2 and f′ is a η-convex function with η bounded from above on f([a,b])×f([a,b]), then(48)fa+fb2∫abgxdx-∫abfxgxdx≤b-a42f′b+ηf′a,f′b∫01∫1+t/2a+1-t/2b1-t/2a+1+t/2bgududt.If in Theorem16 we set η(x,y)=x-y and g≡1, then we have the following result.Using of Theorems15 and 16 can result in some numerical inequalities. This fact is shown in the following example.Example 17.
Consider all assumptions of Example11 and g(x)=(1-x)2(x≥0) which is symmetric to x=1. From inequality (46) in Theorem 15, for 0≤a<1<b≤2 with a+b=2, we get(49)∫abFxx-12dx-F1∫abx-12dx≤Kb-a∫1b2xb-a+a2-b2x-12dx,where (50)K=min⁡2a+1,a+32.
Further calculations in (49) show that (51)15b5-a5-12b4-a4+b2-a2-b-a≤Kb-14, where (52)K=2a+1,a≥13;a+32,a<13.As an application of Theorems15 and 16, we give an error estimate for trapezoidal and midpoint formula that are generalization of Proposition 4.1 in [17] and Proposition 4.1 in [18], respectively.Suppose thatd is a partition a=x0<x1<⋯<xn-1<xn=b of interval [a,b]. Consider quadrature formula (53)∫abfxgxdx=Tjf,g,d+Ejf,g,d,j=1,2,where (54)T1f,g,d=∑i=0n-1fxi+fxi+12∫xixi+1gxdx,T2f,g,d=∑i=0n-1fxi+xi+12∫xixi+1gxdx and Ej(f,g,d) is the approximation error for trapezoidal and midpoint formula.Theorem 18.
Suppose thatf:[a,b]→R is a differentiable function, g:[a,b]→R+ is a continuous function and symmetric with respect to a+b/2, and f′ is a η-convex function where η is bounded from above on [a,b]. Then(55)E1f,g,d≤∑i=0n-1xi+1-xi42f′xi+1+ηf′xi,f′xi+1×∫01∫1+t/2xi+1-t/2xi+11-t/2xi+1+t/2xi+1gxdxdt.Proof.
It is enough to apply Theorem15 on the subinterval [xi,xi+1] (i=0,1,…,n-1) of the partition d for interval [a,b] and to sum all achieved inequalities over i and then use triangle inequality.Theorem 19.
Suppose thatf:[a,b]→R is a differentiable mapping, g:[a,b]→R+ is a continuous mapping symmetric about a+b/2, and f′ is a η-convex mapping on [a,b] with a bounded η from above. Then(56)E2f,g,d≤∑i=0n-11xi+1-xi∫xi+xi+1/2xi+1x-xi2-xi+1-x2gxKidx,where (57)Ki=min⁡f′xi+1+ηf′xi,f′xi+12,f′xi+ηf′xi+1,f′xi2,for i=0,1,…,n-1.Proof.
It is enough to apply Theorem16 on the subinterval [xi,xi+1] (i=0,1,…,n-1) of the partition d for interval [a,b] and to sum all achieved inequalities over i and then use triangle inequality.Example 20.
Suppose thatd is the partition x0=0<x1=1<x2=2 of interval [0,2]. Consider the functions F,g and the bifunction η as defined in Example 17 for a=0,b=2. So the approximation error related to the trapezoidal and midpoint formula can be calculated as the following. From (55) we get(58)E1F,g,d≤∑i=01xi+1-xi42F′xi+1+ηF′xi,F′xi+1×∫01∫1+t/2xi+1-t/2xi+11-t/2xi+1+t/2xi+1x-12dxdt=142+η0,1∫01∫1-t/21+t/2x-12dxdt+142+η1,1∫01∫3-t/23+t/2x-12dxdt=132+124∫01t-13+t+13dt=49192. Also from (56) we have (59)E2F,g,d≤∫1/21x2-1-x2x-12K0dx+∫3/22x-12-2-x2x-12K1dx, where (60)K0=min⁡1+η0,12,η0,12,K1=min⁡1+η1,12,1+η1,12.Then with some calculations we get (61)E2F,g,d≤196K0+1796K1=35192. Note that we can decrease the approximation error by choosing n>2.


## 3. Conclusions

The convexity of a function is the basis for many inequalities in mathematics. Note that, in new problems related to the convexity, generalized notions about convex functions are required to obtain applicable results. One of this generalizations is the notion ofη-convex functions which can generalize many inequalities related to convex functions such as Hermite-Hadamard inequality, Fejér inequality, and trapezoid and midpoint type inequality.



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