Reversed Hardy inequality for C-monotone functions

Abstract

Abstract. In this paper, we will give general Hardy and reversed Hardy type inequalities for a generalized class of monotone functions. Moreover we will give n -exponential convexity, exponential convexity and related results for some functionals obtained from the differences of these inequalities. At the end we will give mean value theorems and Cauchy means for these functionals.Mathematics subject classification (2010): 26D15, 26A48.

New generalizations of Popoviciu-type inequalities via new Green’s functions and Montgomery identity

“…There are remarkable changes in the results, which force to think about the existence of such results. Butt and Pečarić paid tribute to Professor T. Popoviciu in their book [ 3 ] in 2015 to commemorate 50 years to Popoviciu’s inequality. They generalized Popoviciu’s inequality for higher order convex functions and gave its applications.…”

Weighted averages of n-convex functions via extension of Montgomery’s identity

“…is an absolutely continuous function with h (n+1) ≥ 0 and let λ ∈ [a, b] m and ρ ∈ R m satisfy m i=1 ρ i = 0 and m i=1 ρ i λ i = 0. Then representations (3.4) and (3.5) hold and the remainders R j n (h; a, b), j = 1, 2, satisfy the bounds 2) (a) , (3.9) using (3.9) and the identities from Theorem 2.1, we deduce (3.8).…”

“…Remark 3.9 Left-hand sides of the inequalities (2.8), (2.10), (2.15) and (2.18) can be defined as linear functionals in h. Using similar methods as in [2] we can prove mean value results for these functionals, as well as construct new families of exponentially convex functions and Cauchy-type means. Then, using some known properties of exponentially convex functions, we can derive new inequalities and prove monotonicity of the obtained Cauchy-type means analogously as in [2].…”

New Estimates for Hermite-Hadamard Inequality in Quantum Calculus via (α, m) Convexity

“…Many researchers have studied integral inequalities in classical calculus along with their applications (see [6][7][8][9]) . Because the value of mathematical inequalities was well established in past, inequalities such as Hermite-Hadamard, Popoviciu's, Steffensen-Grüss, Jensen, Hardy and Cauchy-Schwarz performed an essential role in the theory of classical calculus and q-calculus [10][11][12][13][14].…”