New generalized Pólya–Szegö and Čebyšev type inequalities with general kernel and measure

Abstract

It is always attractive and motivating to acquire the generalizations of known results. In this article, we introduce a new class $\mathfrak{C(h)}$ C ( h ) of functions which can be represented in a form of integral transforms involving general kernel with σ-finite measure. We obtain some new Pólya–Szegö and Čebyšev type inequalities as generalizations to the previously proved ones for different fractional integrals including fractional integral of a function with respect to another function capturing Riemann–Liouville integrals, Hadamard fractional integrals, Katugampola fractional integral operators, and conformable fractional integrals. This new idea shall motivate the researchers to prove the results over a measure space with general kernels instead of special kernels.

Some New Beesack–Wirtinger-Type Inequalities Pertaining to Different Kinds of Convex Functions

“…Ayub et al in [15] used new a Mittag-Leffler function and derived its applications. Iqbal et al in [16] found new generalized Pólya-Szegö-and Chebyshev-type inequalities with a general kernel and measure. Gul et al in [17] investigated a class of boundary-value problems under the ABC fractional derivative.…”