Some bounds for alternating Mathieu type series

Abstract

Using recent investigated integral representations for the generalized alternating Math- ieu series ˜ S (α,β) μ � r; {a n} ∞=1 �� r,α,β,μ, {a n} ∞=1 ∈ R + � (9,14,18) with a n = nγ, γ ∈ R+ and Mellin-Laplace type integral transforms for the generalized hypergeometric functions and the Bessel function offirstkind, somebounding inequalities for ˜ S (α,β) μ � r; {n γ } ∞ n=1 � are presented. Namely, it is shown that the series ˜ S (α,β) μ � r; {n γ } ∞ n=1 � under some conditions for parameters α, β, γ and μ are bounded with constants which do not depend on α ,β and γ but only depend on r and μ,i.e. ˜ S ( α,β) μ � r; � n γ � ∞=1 � 2 (1 + r2) μ .