Simple sufficient conditions for starlikeness and convexity for meromorphic functions

Abstract In this paper we investigate some extensions of sufficient conditions for meromorphic multivalent functions in the open unit disk to be meromorphic multivalent starlike and convex of order α. Our results unify and extend some starlikeness and convexity conditions for meromorphic multivalent functions obtained by Xu et al. [2], and some interesting special cases are given.


Introduction and definitions
Let † p;n denote the class meormorphic multivalent functions of the form f .z/ D z p C 1 X kDn a k p z k p .p; n 2 N WD f1; 2; 3 : : : g/ ; (1) which are analytic in the punctured unit disk U WD U n f0g, where U WD fz 2 C W jzj < 1g. (ii) Further, we denote by K n the class of n-convex functions in U, i.e. f 2 A n and satisfies Re Â 1 C zf 00 .z/ f 0 .z/ Ã > 0; z 2 U: In the recent papers of Goyal et al. [1] and Xu et al. [2], the authors obtained some sufficient conditions for multivalent and meromorphic starlikeness and convexity, respectively. In this paper we will derive some extensions of these sufficient conditions for starlikeness and convexity of order˛for meromorphic multivalent functions.

Main results
In order to find some simple sufficient conditions for the starlikeness and convexity of order˛for a function f 2 † p;n , we will recall the following lemma due to P. T. Mocanu (see also [3]): ). If f 2 A n and satisfies the inequality Remark that, for the special case n D 1, this result was previously obtained in [5,Theorem 3].
2. If f 2 † p;n , with f .z/ ¤ 0 for all z 2 U , satisfies the inequality for some real values of˛.0 Ä˛< p/, then f 2 †S p;n .˛/. (The power is the principal one).
Proof. For f 2 † p;n , with f .z/ ¤ 0 for all z 2 U , let us define a function h by Since f 2 † p;n and f .z/ ¤ 0 for all z 2 U , it follows that the power function If the function f is of the form (1), a simple computation shows that h.z/ D z C a n p p z nC1 C : : : ; z 2 U. Thus h 2 A n . Now, differentiating logarthmically the definition relation (3) we obtain that which givesˇh From the above relation, by using the assumption (2) of the theorem we geť hence, according to Lemma 2.1, we deduce that h 2 S n . Using again (4), we get and according to the fact that h 2 S n , the above relation implies If h satisfies the inequalityˇh then f 2 †S p;n .˛/. (The power is the principal one).
Proof. As in the proof of Theorem 2.2, we have h 2 A n . Moreover, from the assumption (6) we deduce thať Therefore, the function h satisfies the condition of Lemma 2.1, and thus h 2 S n . Now, using the same reasons as in the last part of the proof of Theorem 2.2, we finally obtain that f 2 †S p;n .˛/: Next, we will give some sufficient conditions for a function f 2 † p;n to be a convex function of order˛.
for some real values of˛.0 Ä˛< p/, then f 2 †K p;n .˛/. (The power is the principal one).
Proof. For f 2 † p;n with f 0 .z/ ¤ 0 for all z 2 U , the power function has an analytic branch in U with '.0/ D 1, and if the function f is of the form (1), then '.z/ D 1 n p p.˛ p/ a n p z n C : : : ; z 2 U: It follows that the function h defined by '.t /dt D z n p p.˛ p/.n C 1/ a n p z nC1 C : : : ; z 2 U; belongs to A n . Thus, we deduce that the function g defined by g.z/ D zh 0 .z/ D z n p p.˛ p/ a n p z nC1 C : : : ; z 2 U; is in A n . From the above definition relation, we get From here and using the assumption (7), we obtaiň Therefore, from Lemma 2.1 it follows that g.z/ D zh 0 .z/ 2 S n , which is equivalent to h 2 K n . Noting that hence f 2 †K p;n .˛/.
Theorem 2.5. If f 2 † p;n , with f 0 .z/ ¤ 0 for all z 2 U , satisfies the inequality for some real values of˛.0 Ä˛< p/, then f 2 †K p;n .˛/. (The power is the principal one).
Proof. For f 2 † p;n with f 0 .z/ ¤ 0 for all z 2 U , the function ' defined by (8) is in A n , therefore the function h given by (9) is in A n . Further, letting g.z/ D zh 0 .z/, we obtain that g.z/ D z n p p.˛ p/ a n p z nC1 C 2 A n ; 00 . e iÂ /ˇd Cˇzh 00 .z/ˇ; z 2 U: Since using the assumption (10) we geť and from (11), using again (10) we deduce thať According to Lemma 2.1 we obtain that g.z/ D zh 0 .z/ 2 S n , which is equivalent to h 2 K n . Consequently, as in the last part of the proof of Theorem 2.4 it follows that f 2 †K p;n .˛/.
(i) If we put n D 1 in Theorem 2.2 and Theorem 2.3, we get the results established by Xu et al. [2].
(ii) For the special case n D 1, Theorem 2.4 and Theorem 2.5 represent the results of Xu et al. [2].
For f 2 † p;n , with f .z/ ¤ 0 for all z 2 U , let's define the function F by where the power is the principal one. Thus, F .z/ D z C n C 1 a n p z nC1 C 2 A n , and considering this integral operator we derive the next result: Theorem 2.7. If f 2 † p;n , with f .z/ ¤ 0 for all z 2 U , satisfies the inequality The power is the principal one).
Proof. If f 2 † p;n , with f .z/ ¤ 0 for all z 2 U , then Defining the function g.z/ D zF 0 .z/, it follows that g 2 A n , anď g 0 .z/ 1ˇDˇF 0 .z/ C zF 00 .z/ 1ˇÄˇF 0 .z/ 1ˇCˇzF 00 .z/Ďˇz From (13), using the assumption (12) we geť and the inequality (14) combined again with (12) implies thať Consequently, from Lemma 2.1 we obtain that g.z/ D zF 0 .z/ 2 S n , which is equivalent to F 2 K n . Using the fact that F 0 .z/ D OEz p f .z/ , it follows that and since F 2 K n we conclude that f 2 †S p;n Let's consider the function f defined by where 0 Ä˛< p, the power is the principal one, and assuming that the parameter 2 C is chosen such that Using MAPLE™ software, from Figure 1a we may see that max jzjÄ1ˇ1 sin z zˇ< 0:18; therefore (16) holds whenever j j Ä 50 9 D 5:555 : : : , and consequently, if 2 C satisfies this inequality then f 2 † p;2 .