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Motivated by several generalizations of the well-known Mathieu series, the main object of this paper is to introduce new extension of generalized Mathieu series and to derive various integral representations of such series. Finally, master bounding inequality is established using the newly derived integral expression.

The main focus of the present paper is to establish definite integral formulae for ratios of the Fox-Wright functions. As consequences of the master formula, some novel integral formulae are derived for ratios of other special functions which are associated to Fox-Wright Ψ function, like generalized hypergeometric function, modified Bessel function of the first kind and Mittag-Leffler type functions of two and three parameters. Moreover, closed integral form expressions are obtained for a family of Mathieu-type series and for the associated alternating versions whose terms contain the incomplete Fox-Wright function. As applications, functional bounding inequalities are established for the aforementioned series.

The main aim of this short note is to obtain new very general upper bounds for multiple generalized Mathieu series considering the related integral representation obtained recently by Pogány and Tomovski [?], by means of the multiple Hardy-Hilbert type integral inequality given in [?].