Abstract
We investigate some subclasses of -uniformly convex and -uniformly starlike functions in open unit disc, which is generalization of class of convex and starlike functions. Some coefficient inequalities, a distortion theorem, the radii of close-to-convexity, and starlikeness and convexity for these classes of functions are studied. The behavior of these classes under a certain modified convolution operator is also discussed.
1. Introduction
Let be the class of all analytic functions in open unit disc , normalized by and Thus, any has the following Maclaurin’s series:
A function is said to be univalent if it never takes same value twice. By we mean the subclass of which is composed of univalent functions. By and we mean the well-known subclasses of that are, respectively, starlike and convex.
In 1991, Goodman [1, 2] introduced the classes and of uniformly convex and uniformly starlike functions, respectively. A function is uniformly convex if maps every circular arc contained in with center onto a convex arc. The function is uniformly starlike if maps every circular arc contained in with center onto a starlike arc with respect to . A more useful representation of and was given in [3–6] as
In 1999, for , Kanas and Wisniowska [7] introduced the class and as
Observe that , and , .
For fixed , these classes have a nice geometrical representation; for detail see [7–9].
A lot of authors obtain very useful properties of and and their generalization in several direction; for example, see [1, 2, 7, 8, 10, 11] and reference cited therein.
For , in [4] (see also [12]), Ronning introduced the following two important subclasses and as
Recently in [13] El-Ashwah et al. introduced two important subclass and of -uniformly convex starlike functions aswhere and .
Let be defined bythen the modified Hadmard product of and is defined by
We denote by subclass of consisting of functions having all negative coefficients in their Maclaurin’s series expansions, so any has a series of the form:
Let be the class of functions given in (1) for which , . Note that [11].
In recent years, more and more researchers are interested in the above defined classes (see [9, 11, 14–22]).
In this paper, by taking inspiration from the above cited paper, we introduce some new subclasses of analytic functions and obtain some interesting results.
Definition 1. For , , , and , a function is in class if and only if
Also
It is worth mentioning that, for special values of parameters, these classes were extensively studied by many authors; here we mention few of them.(1) [21].(2) [21].(3) [11].(4) [23].(5) [4].(6) [13].
Throughout the paper , , , and , unless otherwise stated.
2. Main Results
Theorem 2. A function given by (1) is in class ifwhere
Proof. It is sufficient to prove that inequality (9) holds. As we knowthen inequality (9) can be written as This is, where then we haveNowand alsoFrom (18) and (19), we have The last expression is bounded below by if which completes the proof.
In the next theorem, we prove that condition (11) is also necessary for function .
Theorem 3. Let be given by (1) and in ; then if and only if
Proof. From Theorem 2, we need only to show that satisfies inequality (22). If , then by definition, we have Since is function of form (1) with the argument property given in class and letting in the above inequality, we havefor , and (24) leads to require inequality The functionis extremal function.
Corollary 4. Let given in (1) be in class . Then
Inequality (27) is attained for the function given in (26).
Theorem 5. Let the function given in (1) be in class Then for
The results in (28) are attained for the function given in (26) for .
Proof. As we know from Theorem 3Assimilarly This completes the proof.
Theorem 6. Let the function given in (1) be in class Then for
Proof. For given by (1), we haveIn view of Theorem 3,or, equivalently,A substitution from (35) into (33) yields inequality (32), which is required.
Theorem 7. Let with argument property as in class . Define andwhere , .
Then function is in class if and only if it can be expressed aswhere and .
Proof. Assume that Then it follows thatby Theorem 3, . Conversely, assume that the function defined by (1) belongs to class , and then Set and , . Then and this completes the proof.
Theorem 8. Let . Then is close to convex of order in the disc , where
Proof. As , where is close to convex of order , we haveas this expression is less than if By the fact that if and only if inequality (43) is true if or, equivalently,
Theorem 9. Let . Then is close to convex of order in the , where
Proof. As and is starlike of order , then we haveas The last expression is less than if Using the fact that if and only if (50) is true if Or equivalentlywhich is required.
Theorem 10. Let . Then is convex of order in the , where
Proof. Using the fact that is convex if and only if is starlike, following the lines of Theorem 9, we have the required results.
Theorem 11. Let given by (6) be in class . Then , for
Proof. We need to prove the largest such that From Theorem 3, we have By Cauchy-Schwarz inequality, we haveThus, it is sufficient to showFor Note thatWe need to showor equivalently is an increasing function for For in (65),which proves main assertion of Theorem 11.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors jointly work on the results, and they read and approved the final manuscript.
Acknowledgments
The work here is supported by MOHE Grant: FRGS/1/2016/STG06/UKM/01/1.