Tom 27 (2019): Zeszyt 2 (December 2019)

In the present investigation, by making use of strong differential subordinations and superordinations, we introduce and study two new classes of holomorphic functions containing generalized differential operator. Also we determine important properties for functions belongs to these classes.

In this scientific note, an operator, which is the well-known Tremblay operator in the literature, is first introduced and some of its applications to certain analytic complex functions, which are normalized and analytic in the open unit disk, are then determined. In addition, certain special results of the related applications are also emphasized.

In the present paper we establish some theorems on the existence of common approximate fixed points for a pair of generalized contractive type mappings with the property that the diameter of the set of common ∈−fixed points is tending to zero as ∈ tends to zero.

In this paper we have introduced two new types of sets termed as 𝕀*_µ sets and strongly 𝕀*_µ -open sets and discussed some of its properties. The relation between similar types of sets, characterizations and some basic properties of such sets have been studied.

In this paper we introduce a new general integral operator for analytic functions in the open unit disk 𝕌 and we obtain sufficient conditions for univalence of this integral operator.

A new subfamily of p–valent analytic functions with negative coefficients in terms of q–analogue of generalized Ruschweyh operator is considered. Several properties concerning coefficient bounds, weighted and arithmetic mean, radii of starlikeness, convexity and close-to-convexity are obtained. A family of class preserving integral operators and integral representation are also indicated.

In this paper an improved error bound is obtained for the complete quartic spline with deficiency 2, in the less smooth class of continuous functions. In the case of Lipschitzian functions, the obtained estimate improves the constant from Theorem 3, in J. Approx. Theory 58 (1989) 58-67. Some applications of the complete quartic spline in the numerical integration and in the construction of an iterative numerical method for fourth order two-point boundary value problems with pantograph type delay are presented.

In this paper we obtain estimations of the errors in approximation by positive linear operators which fix certain functions. We use both the first and the second order classical moduli of smoothness and a generalized modulus of continuity of order two. Some applications involving Bernstein type operators, Kantorovich type operators and genuine Bernstein-Durrmeyer type operators are presented.

We consider classical entropies associated with several continuous distributions of probabilities. Explicit expressions and properties of them are presented.