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Research article

Several integral inequalities for (α, s,m)-convex functions

  • Received: 11 December 2019 Accepted: 03 April 2020 Published: 26 April 2020
  • MSC : 26A51, 26D15

  • In this paper, we establish several new integral inequalities for (α, s, m)-convex functions. We recapture the Hermite-Hadamard inequality as a particular case. In order to obtain our results, we use classical inequalities such as Hölder inequality, Hölder-Işcan inequality and Power mean inequality. We formulate several bounds involving special functions like classical Euler-Gamma, Beta and PsiGamma functions. We also give some applications.

    Citation: M. Emin Özdemir, Saad I. Butt, Bahtiyar Bayraktar, Jamshed Nasir. Several integral inequalities for (α, s,m)-convex functions[J]. AIMS Mathematics, 2020, 5(4): 3906-3921. doi: 10.3934/math.2020253

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  • In this paper, we establish several new integral inequalities for (α, s, m)-convex functions. We recapture the Hermite-Hadamard inequality as a particular case. In order to obtain our results, we use classical inequalities such as Hölder inequality, Hölder-Işcan inequality and Power mean inequality. We formulate several bounds involving special functions like classical Euler-Gamma, Beta and PsiGamma functions. We also give some applications.


    Many research papers have studied the properties of convex functions that make this concept interesting in mathematical analysis [1,2,3,4]. In recent years, important generalizations have been made in the context of convexity: quasi-convex [5], pseudo-convex [6], invex and preinvex [7], strongly convex [8], approximately convex [9], MT-convex [10], (α,m)-convex [11], and strongly (s,m)-convex [12,13,14,15]. Here, we recall the notion of convexity: A function g:[β1,β2]RR is said to be convex if the following inequality holds

    g(tx+(1t)y)tg(x)+(1t)g(y),x,y[β1,β2],t[0,1]. (1.1)

    Now, we recall our basic definition, the so-called quasi-convex function.

    Definition 1.1 ([16]). A function g:[β1,β2]R is said quasi-convex on [β1,β2] if

    g(tx+(1t)y)max{g(x),g(y)}, (1.2)

    for any x,y[β1,β2] and t[0,1].

    It is important to note that, any convex function is a quasi-convex but the reverse is not true. In the following example we explain that fact.

    Example 1.1 ([5]). The function h:[2,2]R, defined by

    h(s)={1, for s[2,1],s2, for s(1,2],

    is not convex on [2,2] but it is easy to see that the function is quasi-convex on [2,2].

    Notice that h is quasi-convex if and only if all the level sets of h are intervals (convex sets of the line).

    The use of the convex function to study the integral inequalities have been deeply investigated, especially for the well-known inequality of Hermite-Hadamard type (HH-type inequality). The HH-type inequalities are one of the most important type inequalities and have a strong relationship to convex functions. In 1893 Hermite and Hadamard [17] found independently that for any convex function g:[β1,β2]R, the inequality

    g(β1+β22)1β2β1β2β1g(x)dxg(β1)+g(β2)2, (1.3)

    holds.

    In the field of mathematical analysis, many scholars have focused on defining new convexity and implementing of the problems based on their features. The features that make the results different from each other include lower and higher order derivative of the function. The differential equations with impulse perturbations lie in a special significant position in the theory of differential equations. Among them, integral inequality methods are the important tools to investigate the qualitative characteristics of solutions of different kinds of equations such as differential equations, difference equations, partial differential equations, and impulsive differential equations; see [18,19,20,21,22,23] for more details.

    The HH-type inequality (1.3) has been applied to various convex functions like s-geometrically convex functions [24], GA-convex functions [25], MT-convex function [10], (α,m)-convex functions [26] and many other types can be found in [27]. Besides, the HH-type inequality (1.3) has been applied to a numerous type of convex functions in the sense of fractional calculus like F-convex functions [28], λψ-convex functions [29], MT-convex functions [30], (α,m)-convex functions [11], new class of convex functions [31] and many other types can be found in the literature. Meanwhile, it has been applied to other models of fractional calculus like standard Riemann-Liouville fractional operators [32,33], conformable fractional operators [34,35,36], generalized fractional operators [37], ψ-RL-fractional operators [38,39], Tempered fractional operators [40], and AB- and Prabhakar fractional operators [41].

    In view of the above indices, we extend the work done in [42] to establish some modified HH-type inequalities for the 3-times differentiable quasi-convex functions.

    This section deals with our main results. Throughout this paper, we mean gL[β1,β2] that the function g is differential and continuous on [β1,β2].

    Lemma 2.1. Suppose that g:JRR is a differentiable function such that β1,β2J with β1<β2. If gL[β1,β2], then we have

    g(β1+β22)1β2β1β2β1g(x)dx+(β2β1)224g(β1+β22)=(β2β1)396[10t3g(tβ1+β22+(1t)β1)dt+10(t1)3g(tβ2+(1t)β1+β22)dt]. (2.1)

    Proof. By applying integration by parts three times to get

    J1:=10t3g(tβ1+β22+(1t)β1)dt=2β2β1g(β1+β22)6β2β110t2g(tβ1+β22+(1t)β1)dt=2β2β1g(β1+β22)12(β2β1)2g(β1+β22)+24(β2β1)210tg(tβ1+β22+(1t)β1)dt=2β2β1g(β1+β22)12(β2β1)2g(β1+β22)+48(β2β1)3g(β1+β22)48(β2β1)310g(tβ1+β22+(1t)β1)dt. (2.2)

    Making use of change of the variable x=tβ1+β22+(1t)β1 for t[0,1] and multiplying by (β2β1)396 on both sides, we obtain

    (β2β1)396J1=(β2β1)248g(β1+β22)β2β18g(β1+β22)+12g(β1+β22)1β2β1β1+β22β1g(x)dx. (2.3)

    Analogously, we can deduce

    (β2β1)396J2:=(β2β1)39610(t1)3g(tβ2+(1t)β1+β22)dt=(β2β1)248g(β1+β22)+β2β18g(β1+β22)+12g(β1+β22)1β2β1β2β1+β22g(x)dx. (2.4)

    Finally, by adding (2.3) and (2.4), we get the required identity (2.1).

    Remark 2.1. Notice that f being quasi-convex is not equivalent to |f| being quasi-convex. For instance, g(x)=x21 is only quasi-convex (but not |g(x)|), whereas g(x)=1 if x[1,1] and g(x)=1 otherwise, is not quasi-convex, but |g(x)|=1 is quasi-convex.

    Theorem 2.1. Suppose that g:J[0,+)R is a differentiable function such that gL[β1,β2], where β1,β2J with β1<β2. If |g| is quasi-convex function on [β1,β2], then we have

    |g(β1+β22)1β2β1β2β1g(x)dx+(β2β1)224g(β1+β22)|(β2β1)3384K, (2.5)

    where K=max{|g(β1+β22)|,|g(β1)|}+max{|g(β1+β22)|,|g(β2)|}.

    Proof. Making use of Lemma 2.1 and the quasi-convexity |g|, we have that

    |g(β1+β22)1β2β1β2β1g(x)dx+(β2β1)224g(β1+β22)|(β2β1)396[10t3|g(tβ1+β22+(1t)β1)|dt+10(t1)3|g(tβ2+(1t)β1+β22)|dt](β2β1)39610t3max{|g(β1+β22)|,|g(β1)|}dt+(β2β1)39610(1t)3max{|g(β1+β22)|,|g(β2)|}dt(β2β1)3384[max{|g(β1+β22)|,|g(β1)|}+max{|g(β1+β22)|,|g(β2)|}].

    This rearranges to the desired result.

    Example 2.1. To clarify the following expression occurs in Theorem 2.1

    p:=(β2β1)224g(β1+β22), (2.6)

    we consider the function g(x)=xx2+2 on the interval [β1,β2]=[0,1]. Then, we have

    y(x):=g(x)=2x(4x23)(x2+2)2;p=124g(12)=462187.

    Figure 1 demonstrates the intersections and relationships between the functions g(x),y(x) and the point p geometrically.

    Figure 1.  Plot illustration for the expression (2.6).

    Corollary 2.1. Let the assumptions of Theorem 2.1 be valid and let

    H:=|g(β1+β22)1β2β1β2β1g(x)dx+(β2β1)224g(β1+β22)|.

    Then,

    (i) if |g| is increasing, then we have

    H(β2β1)3384[|g(β2)|+|g(β1+β22)|], (2.7)

    (ii) if |g| is decreasing, then we have

    H(β2β1)3384[|g(β1)|+|g(β1+β22)|], (2.8)

    (iii) if g(β1+β22)=0, then we have

    H(β2β1)3384[|g(β1)|+|g(β2)|], (2.9)

    (iv) if g(β1)=g(β2)=0, then we have

    H(β2β1)3384|g(β1+β22)|. (2.10)

    Theorem 2.2. Suppose that g:J[0,+)R is a differentiable function such that gL[β1,β2], where β1,β2J with β1<β2. If |g|q is quasi-convex function on [β1,β2] and q>1 with 1p+1q=1, then we have

    |g(β1+β22)1β2β1β2β1g(x)dx+(β2β1)224g(β1+β22)|(β2β1)396(3p+1)1pKq, (2.11)

    where Kq=(max{|g(β1+β22)|q,|g(β1)|q})1q+(max{|g(β1+β22)|q,|g(β2)|q})1q.

    Proof. Let p>1. Then from Lemma 2.1 and using the Hölder inequality, we can deduce

    |g(β1+β22)1β2β1β2β1g(x)dx+(β2β1)224g(β1+β22)|(β2β1)396[10t3|g(tβ1+β22+(1t)β1)|dt+10(1t)3|g(tβ2+(1t)β1+β22)|dt](β2β1)396(10t3pdt)1p(10|g(tβ1+β22+(1t)β1)|qdt)1q+(β2β1)396(10(1t)3pdt)1p(10|g(tb+(1t)β1+β22)|qdt)1q.

    The quasi-convexity of |g|q on [β1,β2] implies that

    10|g(tβ1+β22+(1t)β1)|qdtmax{|g(β1+β22)|q,|g(β1)|q},

    and

    10|g(tb+(1t)β1+β22)|qdtmax{|g(β1+β22)|q,|g(β2)|q}.

    Therefore, we obtain

    |g(β1+β22)1β2β1β2β1g(x)dx+(β2β1)224g(β1+β22)|(β2β1)396(3p+1)1pKq,

    where we used the identities

    10t3pdt=10(1t)3pdt=13p+1.

    Thus, our proof is completely done.

    Corollary 2.2. Let the assumptions of Theorem 2.2 be valid and let

    H=|g(β1+β22)1β2β1β2β1g(x)dx+(β2β1)224g(β1+β22)|.

    Then,

    (i) if |g| is increasing, then we have

    H(β2β1)396(3p+1)1p[|g(β2)|+|g(β1+β22)|], (2.12)

    (ii) if |g| is decreasing, then we have

    H(β2β1)396(3p+1)1p[|g(β1)|+|g(β1+β22)|], (2.13)

    (iii) if g(β1+β22)=0, then we have

    H(β2β1)396(3p+1)1p[|g(β1)|+|g(β2)|], (2.14)

    (iv) if g(β1)=g(β2)=0, then we have

    H(β2β1)396(3p+1)1p|g(β1+β22)|. (2.15)

    Theorem 2.3. Suppose that g:J[0,+)R is a differentiable function such that gL[β1,β2], where β1,β2J with β1<β2. If |g|q is quasi-convex function on [β1,β2] and q1 with 1p+1q=1, then we have

    |g(β1+β22)1β2β1β2β1g(x)dx+(β2β1)224g(β1+β22)|(β2β1)3384Kq, (2.16)

    where Kq is as before.

    Proof. From Lemma 2.1, properties of modulus, and power mean inequality, we have

    |g(β1+β22)1β2β1β2β1g(x)dx+(β2β1)224g(β1+β22)|(β2β1)396[10t3|g(tβ1+β22+(1t)β1)|dt+10(1t)3|g(tβ2+(1t)β1+β22)|dt](β2β1)396(10t3dt)1p(10t3|g(tβ1+β22+(1t)β1)|qdt)1q+(β2β1)396(10(1t)3dt)1p(10(1t)3|g(tb+(1t)β1+β22)|qdt)1q.

    Then, by using the quasi-convexity of |g|q on [β1,β2], we have

    10t3|g(tβ1+β22+(1t)β1)|qdt14max{|g(β1+β22)|q,|g(β1)|q},

    and

    10(1t)3|g(tb+(1t)β1+β22)|qdt14max{|g(β1+β22)|q,|g(β2)|q}.

    Therefore, we obtain

    |g(β1+β22)1β2β1β2β1g(x)dx+(β2β1)224g(β1+β22)|(β2β1)3384Kq,

    Hence, our proof is completely done.

    Remark 2.2. If q=pp1 (p>1), the constants of Theorem 2.2 are improved, since 1(3p+1)1p<1.

    The following corollary improves the inequalities (2.7)–(2.10).

    Corollary 2.3. Let the assumptions of Theorem 2.3 be valid and let

    H=|g(β1+β22)1β2β1β2β1g(x)dx+(β2β1)224g(β1+β22)|.

    Then,

    (i) If |g| is increasing, we obtain (2.7).

    (ii) If |g| is decreasing, we obtain (2.8).

    (iii) If g(β1+β22)=0, we obtain (2.9),

    (iv) if g(β1)=g(β2)=0, we obtain (2.10).

    Consider the special means of positive real numbers β1>0 and β2>0, define by:

    ● Arithmetic Mean:

    A(β1,β2)=β1+β22.

    ● Logarithmic mean:

    L(β1,β2)=β2β1ln|β2|ln|β1|, |β1||β2|, β1,β20.

    ● Generalized log-mean:

    Lp(β1,β2)=[β2p+1β1p+1(p+1)(β2β1)]1p, pZ{1,0}, β1β2.

    Remark 3.1. Let 0<α1 and x>0. Then, we consider

    g(x)=xα+3(α+1)(α+2)(α+3),g(x)=xα.

    For each x,y>0 and t[0,1], we see that (tx+(1t)y)αtαxα+(1t)αyα, then we see that g(x) is α-convex function on (0,+) and g(β1+β22)=Aα+3(β1,β2)(α+1)(α+2)(α+3). Furthermore, we have

    1β2β1β2β1g(x)dx=1(α+1)(α+2)(α+3)[βα+42βα+41(α+4)(β2β1)]=1(α+1)(α+2)(α+3)Lα+3α+3(β1,β2).

    Above we used the definition of α-convexity [11]: A function g:[0,r]R,r>0 is said to be α-convex, if the following holds:

    g(tx+(1t)y)tαg(x)+(1tα)g(y),x,y[0,r],t,α[0,1].

    Proposition 3.1. Let 0<α1 andβ1,β2R+ with β1<β2, then we have

    384(α+1)(α+2)(α+3)|Aα+3(β1,β2)Lα+3α+3(β1,β2)+(β2β1)2(α+2)(α+3)24Aα+1(β1,β2)|(β2β1)3[max{g(β1+β22)α,βα1}+max{g(β1+β22)α,βα2}].

    Proof. Since xα is quasi-convex for each x>0 and α(0,1) because every non-decreasing continuous function is also quasi-convex, so the assertion follows from inequality (2.5) with g(x)=xα+3(α+1)(α+2)(α+3).

    Proposition 3.2. Let β1,β2R such that β1<β2 and [β1,β2](0,+), then we have

    |A1(β1,β2)L1(β1,β2)+(β2β1)212A3(β1,β2)|(β2β1)364[β41+g(β1+β22)4].

    Proof. The assertion follows from inequality (2.11) with g(x)=1x, x[β1,β2].

    Proposition 3.3. Let β1,β2R with 0<β1<β2 and nN, k5, then for all q1, we have

    |Ak(β1,β2)Lkk(β1,β2)+(β2β1)2k!24(k2)!Ak2(β1,β2)|k!(β2β1)32k+23(3p+1)1p(k3)![(β1+β2)k3+2k3βk32].

    Proof. Let g(x)=xk, x[β1,β2], kN with k5, then we have

    g(x)=k!(k3)!xk3,

    and it is easy to see that g is an increasing and quasi-convex function. Then, by applying Corollary 2.2(ⅰ) above, we have

    |Ak(β1,β2)Lkk(β1,β2)+(β2β1)2k!24(k2)!Ak2(β1,β2)|(β2β1)396(3p+1)1pK,

    where

    K=k!(k3)![max{|β1+β22|k3,|β1|k3}+max{|β1+β22|k3,|β2|k3}].

    Then, by applying Corollary 2.1(9) to g above, we get

    |Ak(β1,β2)Lkk(β1,β2)+(β2β1)2k!24(k2)!Ak2(β1,β2)|k!(β2β1)32k+23(3p+1)1p(k3)![(β1+β2)k3+2k3βk32],

    and this completes the proof.

    Here, we consider two particular functions.

    ● First, we define g:RR, by g(x)=ex.

    Then, we have g(x)=ex and

    ||g||=supt[β1,β2]|g(t)|=eβ2,

    By applying inequality (2.7) for above ||g||, we can deduce

    |(1+(β2β1)224)eβ1+β221β2β1(eβ2eβ1)|(β2β1)3384||g||=(β2β1)3384eβ2.

    Particularly for β1=0, it follows that

    |(1+β2224)eβ221β2(eβ21)|β23384eβ2, (3.1)

    and for β2=1, it follows that

    |2524ee+1|e384.

    ● Now, we define g:R+R, by g(x)=1x.

    Then, since |g(x)|=6x4 is quasi-convex in [β1,β2]R+ and

    ||g||=supt[β1,y]|g(t)|=6β14, 0<β1<y.

    By applying inequality (2.8) to the function g above, we get

    |2β1+β2lnβ2lnβ1β2β1+(β2β1)212(2β1+β2)3|(β2β1)33842||g||=(β2β1)332β14. (3.2)

    In view of (3.2) and Proposition 2, we can deduce

    |A1(β1,β2)L1(β1,β2)+(β2β1)212A3(β1,β2)|(β2β1)364[β41+g(β1+β22)4](β2β1)332β14.

    For further illustration on the inequalities (3.1) and (3.2), we present some plot examples. Figures 2 and 3 illustrate the inequalities (3.1) and (3.2), respectively.

    Figure 2.  Plot illustration for inequality (3.1).
    Figure 3.  Plot illustration for inequality (3.2).

    Let

    k(β2)=(1+β2224)eβ221β2(eβ21),K(β2)=β32384eβ2,

    and

    h(β1,β2)=2β1+β2lnβ2lnβ1β2β1+(β2β1)212(2β1+β2)3,H(β1,β2)=(β2β1)332β41.

    Furthermore, Figures 4 and 5 show K(β2)k(β2) and D(β1,β2):=H(β1,β2)h(β1,β2), receptively. From Figure 5, we can see that all values of D(β1,β2) are positive which confirms the validity of (3.2).

    Figure 4.  Plot illustration for K(β2)k(β2).
    Figure 5.  Plot illustration for D(β1,β2).

    In this paper we have established new Hermite–Hadamard inequality mainly motivated by Alomari et al in [42] for quasi-convex functions with gC3([β1,β2]) such that gL([β1,β2]) and we give some applications to some special means and for some particular functions. We hope that the ideas used in this paper may inspire interested readers to explore some new applications.

    We believe that our results, this new understanding of Hermite-Hadamard integral inequalities for quasi-convex functions, will be vital information for the future studies of these models of integral inequality. One can obtain the similar results for other kind of convex functions

    We would like to express our special thanks to the editor and referees. Also, the third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

    The authors declare no conflict of interest.



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