Dragomir, Silvestru ORCID: 0000-0003-2902-6805 and Garayev, Mubariz Tapdigoglu (2024) New norm inequalities for Jensen's gap of analytic functions in Banach algebras with applications. Georgian Mathematical Journal. ISSN 1572-9176

Abstract

Assume that h : G → C is analytic on the convex domain G and x € L(B; E, A, Μ), the set of Bochnerintegrable functions on a measurable space (E, A, μ) endowed with a countably-Additive scalar measure μ on a ρ-Algebra A of subsets of E and with values in the Banach algebra B. If the spectrum ρ(x(t)) € G for all t € E and γ € G is taken to be close rectifiable curve in G such that ρ(x(t)) € ins(γ) for all t € E, then, in this paper,we show among others that f(h°x)(μ)dμ(u)-h(fx(u)dμ(u))||≤1/2φfe||xξ-X(v)||(fy|h(e)|/(|e|)-||xξ||(|e|-||x(v)|||de|)d μ(v) where Xξ := fξx(u) dμ(u). Some examples for exponential function in Banach algebras are also given. Applications for discrete inequalities and Hermite-Hadamard-Type inequalities are provided as well.

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