Generalized Close-to-Convexity Related with Bounded Boundary Rotation

The class Pα,m[A, B] consists of functions p, analytic in the open unit disc E with p(0) = 1 and satisfy p(z) = (m/4 + ½) p1(z) – (m/4 – 1/2) p2(z), m ≥ 2, and p1, p2 are subordinate to strongly Janowski function (1+Az/1+Bz)α, α ∈ (0, 1] and −1 ≤ B < A ≤ 1. The class Pα,m[A, B] is used to define Vα,m[A, B] and Tα,m[A, B; 0; B1], B1 ∈ [−1, 0). These classes generalize the concept of bounded boundary rotation and strongly close-to-convexity, respectively. In this paper, we study coefficient bounds, radius problem and several other interesting properties of these functions. Special cases and consequences of main results are also deduced.

Some properties for certain class of bi-univalent functions defined by $ q $-Cătaş operator with bounded boundary rotation

<abstract><p>Throughout the paper, we introduce a new subclass $ \mathcal{H}_{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}\ f(z)$ by using the Bazilevič functions with the idea of bounded boundary rotation and $ q $-analogue Cătaş operator. Also we find the estimate of the coefficients for functions in this class. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward $ (p, q) $-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter $ p $ is obviously redundant.</p></abstract>

Toeplitz matrices whose elements are coefficients of Bazilevič functions

We consider the Toeplitz matrices whose elements are the coefficients of Bazilevič functions and obtain upper bounds for the first four determinants of these Toeplitz matrices. The results presented here are new and noble and the only prior compatible results are the recent publications by Thomas and Halim [1] for the classes of starlike and close-to-convex functions and Radhika et al. [2] for the class of functions with bounded boundary rotation.

Some applications of certain integral operators involving functions

Integral transforms map equations from their original domains into others where manipulations and solutions may be much easier than in original domains. To get back in the original environment, we use the idea of inverse of the integral transform. A function analytic and locally univalent in a given simply connected domain is said to be of bounded boundary rotation if its range has bounded boundary rotation which is defined as the total variation of the direction angle of the tangent to the boundary curve under a complete circuit. \qquad The main objective of the present article is to study some applications of certain integral operators to functions of bounded radius rotation involving Janowski functions. We discuss some inclusion results under certain assumption on parameters involve in operators as well as in related subclasses of analytic functions. Most of these results are best possible. We also relate our findings with the existing literature of the subjects.