Restricted-orientation convexity (O-convexity) is the study of geometric objects whose intersections with lines from some xed set O of orientations are empty or connected. The notion of O-convexity generalizes standard convexity, as well as several other types of nontraditional convexity. We introduce and study O-halfspaces, which are analogs of standard halfspaces in the theory of O-convexity,...

We introduce some new classes of words and permutations characterized by the second difference condition π(i − 1) + π(i + 1) − 2π(i) ≤ k, which we call the k-convexity condition. We demonstrate that for any sized alphabet and convexity parameter k, we may find a generating function which counts k-convex words of length n. We also determine a formula for the number of 0-convex words on any fixed...

Various function theorists have successfully defined and investigated different kinds of analytic functions. The applications such functions played significant roles in geometry theory as a field complex analysis. In this work, therefore, certain subclass univalent is using generalized differential operator we discussed \(TS_{\sigma, \delta} ^{~ \wp} (\vartheta ,\hbar ,\ell )\) with negative co...

Let f 2 C 1 ?1; 1] change its convexity nitely many times in the interval, say s times, at Y s : ?1 < y s < < y 1 < 1. We estimate the degree of simultaneous approximation of f and its derivative by polynomials of degree n, which change convexity exactly at the points Y s , and their derivatives. We show that provided n is suuciently large, depending on the location of the points Y s , the rate...

The aim of the present paper is to consider geometric properties such as starlikeness and convexity of the Cesáro partial sums of certain analytic functions in the open unit disk. By using the Cesáro partial sums, we improve some recent results including the radius of convexity. 1 Introduction Let U := {z : |z| < } be a unit disk in the complex plane C and let H denote the space of all analyti...

Classical convexity can be naturally translated to graphs considering shortest paths, giving rise to the geodesic convexity. Moreover, if we consider induced paths between vertices, we get the so-called monophonic convexity. In this work, we present some results involving monophonic sets.

Some existence results on best approximation are proved without starshaped subset and affine mapping in the set up of q-normed space. First, we consider the closed subset and then weakly compact subsets for said purpose. Our results improve the result of Mukherjee and Som [11] and Jungck and Sessa [7] and some known results [4], [9], [12] are obtained as consequence. To achieve our goal, we hav...