On Starlike Functions of Negative Order Defined by q-Fractional Derivative

In this paper, two new classes of q-starlike functions in an open unit disc are defined and studied by using the q-fractional derivative. The class Sq*˜(α), α∈(−3,1], q∈(0,1) generalizes the class Sq* of q-starlike functions and the class Tq*˜(α), α∈[−1,1], q∈(0,1) comprises the q-starlike univalent functions with negative coefficients. Some basic properties and the behavior of the functions in these classes are examined. The order of starlikeness in the class of convex function is investigated. It provides some interesting connections of newly defined classes with known classes. The mapping property of these classes under the family of q-Bernardi integral operator and its radius of univalence are studied. Additionally, certain coefficient inequalities, the radius of q-convexity, growth and distortion theorem, the covering theorem and some applications of fractional q-calculus for these new classes are investigated, and some interesting special cases are also included.

Further results for two certain subclasses of close-to-convex functions

Let [Formula: see text] be the family of analytic and normalized functions [Formula: see text] in the open unit disc [Formula: see text]. In this paper, we consider the following classes: [Formula: see text] and [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. We show that if [Formula: see text], then [Formula: see text] and [Formula: see text] are greater than [Formula: see text], and if [Formula: see text], then [Formula: see text]. Also, some another interesting properties of the class [Formula: see text] are investigated. Finally, the radius of univalence of 2nd section sum of [Formula: see text] is obtained.

On an extension of Ozaki’s condition

It is known that if {f(z)=z^{p}+\sum_{n=p+1}^{\infty}a_{n}z^{n}} and it is analytic in a convex domain {D\subset\mathbb{C}} and, for some real α, we have {\operatorname{\mathfrak{Re}}\{\exp(i\alpha)f^{(p)}(z)\}>0} , {z\in D} , then {f(z)} is at most p-valent in D. This Ozaki condition is a generalization of the well-known Noshiro–Warschawski univalence condition. In this paper, we consider the radius of univalence of functions {g(z)=z+\sum_{n=1}^{\infty}b_{n}z^{n}} such that {g^{\prime}(z)\prec[(1+z)^{2}/(1-z)^{2}]} and some related problems.

Radius Constants for Functions with the Prescribed Coefficient Bounds

For an analytic univalent functionf(z)=z+∑n=2∞anznin the unit disk, it is well-known thatan≤nforn≥2. But the inequalityan≤ndoes not imply the univalence off. This motivated several authors to determine various radii constants associated with the analytic functions having prescribed coefficient bounds. In this paper, a survey of the related work is presented for analytic and harmonic mappings. In addition, we establish a coefficient inequality for sense-preserving harmonic functions to compute the bounds for the radius of univalence, radius of full starlikeness/convexity of orderα (0≤α<1) for functions with prescribed coefficient bound on the analytic part.

Typically real logharmonic mappings

We consider logharmonic mappings of the formf(z)=z|z| 2βhg¯defined on the unit diskUwhich are typically real. We obtain representation theorems and distortion theorems. We determine the radius of univalence and starlikeness of these mappings. Moreover, we derive a geometric characterization of such mappings.

Close-to-starlike logharmonic mappings

We consider logharmonic mappings of the formf=z|z|2βhg¯defined on the unit discUwhich can be written as the product of a logharmonic mapping with positive real part and a univalent starlike logharmonic mapping. Such mappings will be called close-to-starlike logharmonic mappings. Representation theorems and distortion theorems are obtained. Moreover, we determine the radius of univalence and starlikeness of these mappings.

Univalence for convolutions

The radius of univalence is found for the convolutionf∗gof functionsf∈S(normalized univalent functions) andg∈C(close-to-convex functions). A lower bound for the radius of univalence is also determined whenfandgrange over all ofS. Finally, a characterization ofCprovides an inclusion relationship.

A GENERALIZATION OF CERTAIN CLASS OF ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS

We introduce the subclass $T^*(A,B,n,a)$ ($-1 \le A < B\le 1$, $0 < B \le 1$, $n \ge 0$, and $0\le\alpha <1$) of analytic func;tions with negative coefficients by the operator $D^n$. Coefficient estimates, distortion theorems, closure theorems and radii of close-to-convexety, starlikeness and convexity for the class $T^*(A,B,n,a)$ are determined. We also prove results involving the modified Hadamard product of two functions associated with the class $T^*(A,B,n,a)$. Also we obtain Several interesting distortion theorems for certain fractional operators .of functions in the class $T^*(A,B,n,a)$. Also we obtain class perserving integral operator of the form \[F(z)=\afrc{c+1}{z^c}\int_0^z t^{c-1}f(t) dt, \quad c>-1\] for the class $T^*(A,B,n,a)$. Conversely when $F(z) \in T*(A,B,n,a)$, radius of univalence of $f(z)$ defined by the above equation is obtained.