We give a Riemann-type definition of the double Denjoy integral of Chelidze and Djvarsheishvili using the new concept of CD filtering.

The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and those of the second author for Euler’s constant γ and its alternating analog ln(4/π), and on the other hand the infinite products of the first author for e, and of the second author for π and e . We obtain new double integral and infinite product representations of man...

We first construct a sequence of double smooth Picard singular integral operators which do not have to be positive in general. After giving some useful estimates, we mainly show that it is possible to approximate a function by these operators in statistical sense even though they do not obey the positivity condition of the statistical Korovkin theory.

Let f and F be members of . The function f is said to be subordinate to F, or F is said to be superordinate to f , if there exists a function w analytic in U, with w(0)= 0 and |w(z)| < 1, and such that f (z)= F(w(z)). In such a case, we write f ≺ F or f (z)≺ F(z). If the function F is univalent in U, then f ≺ F if and only if f (0)= F(0) and f (U)⊂ F(U) (cf. [1, 2]). Let φ : C2 → C and let h be...

were constructed in R 2 = {(x, y) : x > 0, y > 0} . In the present paper (in case of λ = 0) using the constructed fundamental solutions, a double layer potentials are defined and investigated. Limiting theorems are proved, and integral equations concerning a denseness of potentials of a double layer are found. MSC: Primary 35J15, 35J70.

The present work is a sequel to the papers [3] and [4], and it aims at introducing and investigating a new generalized double zeta function involving the Riemann, Hurwitz, Hurwitz-Lerch and Barnes double zeta functions as particular cases. We study its properties, integral representations, differential relations, series expansion and discuss the link with known results.

Abstract We study sign changes and non-vanishing of a certain double sequence Fourier coefficients cusp forms half-integral weight.

We obtain subordination and superordination preserving properties for the Saigo type generalized fractional differ-integral operator, defined for multivalent functions in the open unit disk. A differential sandwich-type theorem for these multivalent function, and some consequences are also presented.

Let f and F be members of . The function f is said to be subordinate to F, or F is said to be superordinate to f , if there exists a function w analytic in U, with w(0)= 0 and |w(z)| < 1, and such that f (z)= F(w(z)). In such a case, we write f ≺ F or f (z)≺ F(z). If the function F is univalent in U, then f ≺ F if and only if f (0)= F(0) and f (U)⊂ F(U) (cf. [1, 2]). Let φ : C2 → C and let h be...