Univalent and Starlike Properties for Generalized Struve Function
We derive conditions on the parameters , , and so that the function where is the normalized form of generalized Struve function, belongs to the class Also, some sufficient conditions for the function to be in the class are obtained.
1. Introduction and Preliminaries
Let denote the unit disc in the complex plane and let denote the class of functions which are analytic and of the form and normalized by the conditions and Let denote the class of functions such thatSuppose that and are two analytic functions in , and is univalent in . We say that is subordinate to , written or , if and only if and .
A function belongs to the class of starlike functions of order denoted by ifA subclass of the class of starlike functions denoted by for consists of functions for whichWe also note that . In [1–3] the authors have discussed the coefficient bounds and other extremal properties of the class
For , consider the classFrom , we have the strict inclusion . Very recently, Obradović et al.  have discussed the geometric behaviour of functions in . Also the class has been widely studied by many authors in [4, 6–8].
Let us consider the following second-order linear nonhomogenous differential equation (for more details see [9, 10]):where . The function which is called the generalized Struve function of order is defined as a particular solution of (6) and has the series representation as follows:where stands for the Euler gamma function.
Now, we consider the function defined in terms of the generalized Struve function by the transformation: By using the well-known Pochammer symbol (or the shifted factorial) defined for in terms of the Euler -function, we haveWe obtain the following series representation for the function given by (8):where . Also, this function is analytic on and satisfies the following second-order inhomogeneous differential equation:where
Recently, the class and its generalizations have been widely studied by many authors [4–8, 11]. By applying the admissible function method authors in  have obtained conditions on the triplet , , and such that is in the class , where is the confluent hypergeometric function. In , the authors have derived conditions on the parameters , , and such that the function is in , where is the Gaussian hypergeometric function. Moreover, in [9, 10] Yagmur and Orhan have obtained sufficient conditions for the generalized Struve function to be convex, starlike, and univalent. Most of these results were motivated by the research on geometric properties of Gaussian and confluent hypergeometric function.
Motivated by the above-mentioned works, in this paper we use the method of differential subordination to show that is in the class and also we provide sufficient conditions for the function to be in the class and hence univalent.
To prove our main results, we will need the following lemmas.
Lemma 1 (see ). Let (nonconstant) function be analytic in with , .
If attains its maximum value on the circle at a point , then where is a real number and .
Lemma 2 (see ). If an analytic function has the form and satisfies the conditionthen is univalent in
Lemma 3 (see ). Let be a complex number, , and let be a complex number, , , and a regular function on . Iffor all , then the function is regular and univalent in .
Lemma 4 (see ). Let and let be analytic and univalent on except for those for which . Suppose that satisfies the conditionwhere is finite, , and . If and analytic in , , , and further ifthen in .
Suppose that is analytic in with and . Then the condition (16) reduces to a simple form:whenever , , , and .
2. Main Results
Theorem 5. Let , , and such that ; then
Proof. Let , ; then is analytic in with . Since the function satisfies the differential equation (11) and , satisfies the following inhomogeneous differential equation: Using Lemma 4, we will show that , where . For this, if we letand , then it is sufficient to prove that whenever , , , and is real, we have thatIn the last stage of the inequalities we have used the definition of and shown thatwhenever , is real, and . Hence, .
Choosing in the above theorem we have the following Corollary.
Corollary 6. Let , and be such that ; then
The next results give sufficient conditions for the function to be starlike and univalent in the open unit disc.
Theorem 7. If , and such that andthen the function belongs to the class
Proof. For to be in the class we need to prove that Upon settingwe observe that is analytic in and .
Also Now, since satisfies the inhomogeneous differential equation (11), in terms of we see that satisfies the following equation: where Now, we claim that , . By using Lemma 4 with , , and , we need to show thatwhere is real, , and :where Also, when in Theorem 5 we have that if thenAlso,Now, if we show that , then we haveIn order to establish that we need the following results which state that for and if and only ifIn particular, implies thatWhen , we haveUsing the last inequality, we have and so we have provided thatholds, where is given by (35). And, we observe that the above condition is stated in the theorem. Thus, in and hence belongs to .
Theorem 8. Let , and . If satisfies any one of the following inequalities:then is in and hence is univalent in .
Proof. Define a function by and then is analytic in and Differentiating (46) givesHence, from (46) and (47), we haveNow suppose that there exists such thatand then from Lemma 1, we haveTherefore, letting in each of (48), we obtain thatwhich contradicts our assumption (42)–(45), respectively. Therefore, for all ; then from (46) we havewhich implies is in the class and hence univalent.
Theorem 9. Let , , such that . If satisfies the inequalitythen is univalent in .
Proof. Define a function as follows:We see that is analytic in and . Differentiation of (54) givesNow suppose there exists such thatThen from Lemma 1, we haveNow letting in (55), we have where and . Since we haveThis contradicts the hypothesis and therefore for all . Thus, In view of Lemma 2 it implies that is in and hence univalent.
Theorem 10. Let , be a real number such that and let be a complex number which satisfies the inequalityIf is univalent in and for all , then the functionis univalent in , where the values of the complex powers are taken with their principal values.
Proof. Define a functionThen we have .
AlsoFrom (65), we haveFrom the hypothesis, we have (); then, by the Schwarz Lemma (cf , we obtain thatNow, since is univalent in Using (68), we have So, from (61) we have Applying Lemma 3, we obtain the function defined by (62) which is univalent in .
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work of the first author was supported by the Department of Science and Technology, India, with reference to the Sanction Order no. SR/DST-WOS A/MS-10/2013(G). The work of the second author was supported by the grant given under UGC Minor Research Project F. no. 5599/15 (MRP-SEM/UGC SERO). The work of the third author was supported by the grant given under UGC Minor Research Project, 2014-15/MRP-5591/15(SERO/UGC).
D. J. Wright, “On class of starlike functions,” Compositio Mathematica, vol. 21, pp. 122–124, 1969.View at: Google Scholar
L. A. Aksentev, “Sufficient conditions for univalence of regular functions,” Izvestiya Vysshikh Uchebnykh Zavedenii Matematika, no. 4, pp. 3–7, 1958 (Russian).View at: Google Scholar
H. Orhan and N. Yagmur, “Geometric properties of generalized Struve functions,” in Proceedings of the International Congress in Honour of Professor Hari M. Srivastava, Bursa, Turkey, August 2012.View at: Google Scholar
Z. Nehari, Conformal Mapping, McGraw Hill Book Comp, New York, NY, USA, 1952, (Dover, 1975).