Serdica Mathematical Journal

Volume 48, Numbers 1−2, 2022

C O N T E N T S

A B S T R A C T S

THE SOLUTION TO THE NON-SYMMETRIC COMPLETION PROBLEM OF QUASI-UNIFORM SPACES
Athanasios Andrikopoulos aandriko@ceid.upatras.gr,
John Gounaridis igounaridis@upatras.gr

2020 Mathematics Subject Classification: 54D35, 54E15, 54E25, 54E50, 54E55.
Key words: Quasi-metric, quasi-uniformity, Dedekind-MacNeille completion, cauchy net, embedding, completeness.

In this paper, we give a solution to the non-symmetric completion problem of quasi-uniform spaces. The proposed completion theory extends the uniform space completion theory and is also exceedingly well-behaved in the sense that it satisfies all of the requirements posed in the literature for a nice completion. The main contribution of this completion theory is the concept of the cut of nets, which is a common generalization of Doitchinov's D-Cauchy net and the Eudoxus-Dedekind-MacNeille cut.

A NEWTON--SECANT METHOD FOR DIFFERENTIABLE SET-VALUED MAPS IN NACHI--PENOT SENSE
Michaël Gaydu michael.gaydu@univ-antille.fr,
Olguine Yacinthe yolguine@yahoo.fr,
Silvere Paul Nuiro paul.nuiro@univ-antilles.fr,
Alain Pietrus alain.pietrus@univ-antilles.fr

2020 Mathematics Subject Classification: 49J53, 49J40, 90C48, 65K10.
Key words: Variational inclusions, set-valued maps, divided differences, generalized differentiation of set-valued maps, normed convex processes, majorizing sequences.

This paper is concerned with the solving of variational inclusions of the form \(0\in f(x) + g(x) + F(x) - K\), where \(g\) is a function which is differentiable at a solution \(x^{*}\) of the inclusion but may be not differentiable in a neighborhood of \(x^{*}\). The function \(f\) and the set-valued mapping \(F\) are differentiable in the sense of Nachi--Penot [K. Nachi, J.-P. Penot. Inversion of multifunctions and differential inclusions. Control Cybernet. 34, no. 3 (2005), 871−901] and \(K\) is a nonempty closed convex cone.
We introduce a Newton-Secant method to solve our problem and the sequence associated is semilocally convergent to \(x^{*}\) with an order equal to \(\frac{1 +\sqrt{5}}{2}\). Finally, some numerical results are also given to illustrate the convergence of the proposed method.

2020 Mathematics Subject Classification: 53A05, 53A07, 30F15.
Key words: Generalized Weingarten surfaces, holomorphic functions, Weierstrass-type representation, Appell surfaces.

In this paper we introduce a class of Weingarten hypersurfaces called generalized Weingarten hypersurfaces of radial support type (in short, RSGW hypersurfaces). These hypersurfaces are parameterized using a new technique consisting in obtain locally any hypersurface in Euclidean space as an envelope of a sphere congruence wherein the other envelope is contained in a sphere. We extend the definition of the classical Appell surfaces to hypersurfaces and we characterize both Appell and RSGW hypersurfaces in terms of a same harmonic function in the sphere. For two-dimensional case, we provide a Weierstrass-type representation for RSGW surfaces and from this we get a Weierstrass-type representation for the classical Appell surfaces. These representations depend on two holomorphic functions in such a way that a same pair of functions provides examples in each class. Using it, we give a classification for the rotational cases for both classes. Furthermore is provided a necessary and sufficient condition on the holomorphic data of these classes so that they are parameterized by lines of curvature. We also prove that a compact, complete RSGW surface is a sphere.

CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS INVOLVING HURWITZ-LERCH ZETA FUNCTION
Kishor C. Deshmukh kishord1382@gmail.com,
Rajkumar N. Ingle ingleraju11@gmail.com,
Pinninti Thirupathi Reddy reddypt2@gmail.com

2020 Mathematics Subject Classification: 30C45.
Key words: analytic, starlike, convexity, partial sums, neighborhood.

Making use of Integral operator involving the Hurwitz-Lerch zeta function, we introduce a new subclass of analytic functions defined in the open unit disk and investigate its various characteristics. Further we obtain some usual properties of the geometric function theory such as coefficient bounds, extreme points radius of starlikness and convexity, partial sums and neighbourhood results belonging to the class.

CERTAIN SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS DEFINED BY \(q\)-DIFFERENTIAL OPERATOR
Gajanan M. Birajdar gajanan.birajdar@mitwpu.edu.in,
Naveneet D. Sangle navneet_sangle@rediffmail.com

2020 Mathematics Subject Classification: 30C45, 30C50.
Key words: Harmonic, univalent, Salagean q-differential operator.

In this paper, we define certain subclass of harmonic univalent function in the unit disc \(U = \left\{ {z \in C:\left| z \right| < 1} \right\}\) by using \(q\)-differential operator. Also we obtain coefficient inequalities, growth and distortion theorems for this subclass.