This paper investigates several integral inequalities held simultaneously for q and h-integrals in implicit form. These inequalities are established for symmetric functions using certain types of convex functions. Under certain conditions, Hadamard-type inequalities are deducible for q-integrals. All the results are applicable for ℏ-convex, m-convex and convex functions defined on the non-negative part of the real line.

This paper aims to unify q-derivative/q-integrals and h-derivative/h-integrals into a single definition, called $q-h$-derivative/$q-h$-integral. These notions are further extended on the finite interval $[a,b]$ in the form of left and right $q-h$-derivatives and $q-h$-integrals. Some inequalities for $q-h$-integrals are studied and directly connected with well known results in diverse fields of science and engineering. The theory based on q-derivatives/q-integrals and h-derivatives/h-integrals can be unified using the $q-h$-derivative/$q-h$-integral concept.

In this paper, we give a refinement of the well known Jessen's inequality via weight functions. We discuss \(m\)-exponential convexity of the functions associated with these weighted Jessen's functionals. Cauchy and Lagrange mean value theorems are also given that are useful in the construction of means with Stolarsky property.

The present article establishes some results on Hermite-Hadamard type inequalities of coordinated (p1,h1)-(p2,h2)-convex functions by using Katugampola fractional integrals. The results are in generalized form and deduced for various special cases like coordinated pq-convex functions, coordinated (p,s)-convex functions, coordinated s-convex functions and classical case of coordinated convex functions.

The well-known Hermite-Hadamard inequality for convex functions is extensively studied for different kinds of integrals and derivatives. This paper investigates some of its variants for $ q-h $-integrals using properties of convex functions. Inequalities for $ q $-integrals that have been published in recent years can be extracted from the main results of this paper.

This paper aims to unify q-derivative/q-integral and h-derivative/h-integral in a single definition, this will be called q−h-derivative/q−h-integral. These notions are further extended on finite interval [a,b] in the form of left and right q−h-derivatives and q−h-integrals. Some inequalities are studied for q−h-integrals which are directly connected with well known results. In diverse fields of science and engineering the theory based on q-derivatives/q-integrals and h-derivatives/h-integrals can be unified by using the concept of q−h-derivative/q−h-integral.