GENERALISATION OF HERMITE-HADAMARD TYPE INEQUALITIES FOR (m, α, β, γ, µ) CONVEX FUNCTIONS IN MIXED KINDS AND APPLICATIONS IN NUMERICAL INTEGRATION AND PROBABILITY THEORY
Faraz Mehmood*, Sumaira Yousuf Khan, Faisal Nawaz, Akhmadjon Soleev
Abstract: In the present article, the generalised notion of (m, α, β, γ, µ) −convex(concave) function in mixed kind is introduced 1st time, which is the generalisation of 15. Our fundamental objective is to develop generalised inequalities of Hermite-Hadamard type for functions whose modulus of the derivatives are (m,α,β,γ,µ)−convex by implementing several techniques including H
lder’s & power mean inequalities & applications for the theory of Probability & numerical integration are deduced. Moreover, we deduce more special cases of the class of (m,α,β,γ,µ)−convex function on various choices of m,α,β,γ,µ. Various established consequences of several published articles would be captured as special cases.
Keywords: Convex function, Hermite-Hadamard inequality, Power-mean inequality, Hlder inequality, Beta function, Gamma function, Numerical integration, and Probability density function.