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The main purpose of this current note is to introduce a Hypergeometric distribution series in associated with

integral operator and obtain necessary and sufficient conditions for this integral related series belonging to the classes and T ( , ) and C( , ) .

Suppose that T is a Calderon-Zygmund singular

integral operator.

The invariant subspaces of Volterra

integral operator on the Hilbert Hardy space is studied by Donoghue [8], and a complete characterization of such subspaces in a Banach spaces of analytic functions in the open unit disc, containing the Hardy, Bergman and Dirichlet spaces is obtained in 2008 by Aleman and Korenblum in [4].

22], such an approach is superior to the data-dependent integration of K'[phi], where possible singularities of [phi] as well as the singular kernel of the boundary

integral operator K' have to be treated simultaneously.

Our aim in this paper is to obtain certain fractional integral inequalities for synchronous functions which are related to the Chebyshev functional ([15]) by using the Saigo fractional

integral operator which involves in the kernel, the Gauss hypergeometric function (defined above).

Theorems for factorization and zeros of functions in the space are treated next, followed by

integral operators and equivalent norms, non-conformally invariant space induced by the

integral operator, and Schatten classes of the

integral operator.

mu]] is the Erdelyi-Kober type

integral operator [8].

The Riemann-Liouville fractional

integral operator [J.

Keywords: Analytic functions,

integral operator of the first kind, Bessel function.

3 The Normality and Continuity of The

Integral OperatorWe want to be an

integral operator for the small and medium enterprises (SME) and as such we want to be present in the 3G market and in the battle for the 4G services," CEO Jose Poza told Spanish daily Expansion.