Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex

The authors introduce the concepts of m-invex set, generalized (s, m)-preinvex function, and explicitly (s, m)-preinvex function, provide some properties for the newly introduced functions, and establish new Hadamard-Simpson type integral inequalities for a function of which the power of the absolute of the first derivative is generalized (s, m)-preinvex function. By taking different values of the parameters, Hadamardtype and Simpson-type integral inequalities can be deduced. Furthermore, inequalities obtained in special case present a refinement and improvement of previously known results.

Section: Hadamard-simpson Type Integral Inequalitiesmentioning

confidence: 48%

“…(1.5) Theorem 1.11 ( [2,25]). Let K ⊆ R be an open invex subset with respect to η : K × K → R. Suppose that f : K → R is a differentiable function.…”

Section: Definition 11 ([7]mentioning

confidence: 99%

Properties and integral inequalities of Hadamard- Simpson type for the generalized (s,m) -preinvex functions

Du¹,

Liao²,

Li³

2016

“…The middle term of the inequality in formula (5.14) can be replaced with The type of the Hermite-Hadamard inequality involving the Riemann-Liouville integrals and gamma function were considered in [4], wherein some results were achieved for positive preinvex functions on the open invex set K ⊆ R. The results regarding the Hermite-Hadamard inequality for functions whose absolute values of derivatives are preinvex were obtained in [1].…”

Section: Then the Double Inequalitymentioning

confidence: 99%

Important inequalities for preinvex functions

Pavić¹,

Wu²,

Novoselac³

2016

“…For more results on preinvex and logarithmically preinvex functions see [1,6,10]. In [9], Noor et al presented the Hermite-Hadamard inequalities for logarithmically h-preinvex functions.…”

Section: Introductionmentioning

confidence: 99%

Hermite-Hadamard type inequalities for logarithmically B-preinvex functions

Zafar¹,

Hussain²,

Yasmin³

et al. 2016

In this paper, we introduce the notion of logarithmically B-preinvex functions and establish certain new Hermite-Hadamard type inequalities for the functions whose derivatives in absolute value are logarithmically B-preinvex. Our results generalize several known results for the classes of logarithmically preinvex functions. Some estimates for the left and right hand side of the Hermite-Hadamard inequality are also obtained for a new class of differentiable logarithmically α-preinvex functions.