Research Article A New Class of Meromorphic Functions Associated with Spirallike Functions

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1 Applied Mathematics Volume 202, Article ID 49497, 2 pages doi:0.55/202/49497 Research Article A New Class of Meromorphic Functions Associated with Spirallike Functions Lei Shi, Zhi-Gang Wang, and Jing-Ping Yi School of Mathematics and Statistics, Anyang Normal University, Henan, Anyang , China Correspondence should be addressed to Lei Shi, shilei 04@yahoo.com.cn Received 22 September 202; Revised 26 October 202; Accepted 3 October 202 Academic Editor: Alberto Cabada Copyright q 202 Lei Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a new class of meromorphic functions associated with spirallike functions. Such results as subordination property, integral representation, convolution property, and coefficient inequalities are proved.. Introduction Let Σ denote the class of functions f of the form z a k z k, k 0. which are analytic in the punctured open unit disk U : {z : z C, 0 < z < } : U \ {0}..2 Let P denote the class of functions p given by p p k z k z U,.3 k which are analytic in U and satisfy the condition Re p > 0 z U..4

2 2 Applied Mathematics Let f, g Σ, where f is given by. and g is defined by g z b k z k, k 0.5 then the Hadamard product or convolution f g is defined by f g : z a k b k z k : g f. k 0.6 For two functions f and g,analyticinu, we say that the function f is subordinate to g in U and write g z U,.7 if there exists a Schwarz function ω, which is analytic in U with ω 0 0, ω < z U,.8 such that g ω z U..9 Indeed, it is known that g z U f 0 g 0, f U g U..0 Furthermore, if the function g is univalent in U, then we have the following equivalence: g z U f 0 g 0, f U g U.. A function f Σ is said to be in the class MS β of meromorphic starlike functions of order β if it satisfies the inequality zf Re < β z U; 0 β<..2 For the real number β 0 <β<, we know that zf 2β < 2β zf Re < β..3

3 Applied Mathematics 3 If the complex number satisfies the condition 2 < 2,.4 it can be easily verified that zf 2 < 2 Re zf >..5 We now introduce and investigate the following class of meromorphic functions. Definition.. A function f Σ is said to be in the class MS if it satisfies the inequality Re zf > z U; 2 <..6 2 Remark.2. For 0 < <, the class MS is the familiar class of meromorphic starlike functions of order. Remark.3. If e iψ π/2 <ψ<π/2, then the condition.6 is equivalent to Re e iψ zf < z U,.7 which implies that f belongs to the class of meromorphic spirallike functions. Thus, the class of meromorphic spirallike functions is a special case of the class MS. For some recent investigations on spirallike functions and related functions, see, for example, the earlier works 9 and the references cited in each of these earlier investigations. Remark.4. The function z z 2 Re / z U ; 2 < 2.8 belongs to the class MS. It is clear that Re > 2 <..9 2

4 4 Applied Mathematics Then, for the function f given by.8, we know that Re zf Re > Re [ 2 Re Re ] z z,.20 which implies that f MS. In this paper, we aim at deriving the subordination property, integral representation, convolution property, and coefficient inequalities of the function class MS. 2. Preliminary Results In order to derive our main results, we need the following lemmas. Lemma 2.. Let λ be a complex number. Suppose also that the sequence {A k } k 0 is defined by A 0 2λ, A k 2λ k 2 k A l k N 0 : N {0}. 2. Then A k k! k 2λ j j 0 k N Proof. From 2., we know that k k 2 A k 2λ A l, k k A k 2λ A l. 2.3 By virtue of 2.3, wefindthat A k A k k 2λ k 2 k N Thus, for k, we deduce from 2.4 that A k A k A3 A2 A A 0 A k A 2 A A 0 k! k 2λ j. 2.5 j 0 By virtue of 2. and 2.5, we get the desired assertion 2.2 of Lemma 2..

5 Applied Mathematics 5 Lemma 2.2 Jack s Lemma 0. Let φ be a nonconstant regular function in U. If φ attains its maximum value on the circle z r< at z 0,then z 0 φ z 0 tφ z 0, 2.6 for some real number t t. 3. Main Results We begin by deriving the following subordination property of functions belonging to the class MS. Theorem 3.. A function f MS if and only if zf [ 2 Re ] z z z U ; 2 < Proof. Suppose that h : / zf / i Im / Re / z U; f MS. 3.2 We easily know that h P, which implies that / zf / i Im / Re / ω ω z U; f MS, 3.3 where ω is analytic in U with ω 0 0and ω < z U. It follows from 3.3 that zf [ 2 Re ] ω ω z U, 3.4 which is equivalent to the subordination relationship 3.. On the other hand, the above deductive process can be converse. The proof of Theorem 3. is thus completed. Theorem 3.2. Let f MS.Then [ 2 z exp Re ] z 0 ω t t ω t dt z U, 3.5 where ω is analytic in U with ω 0 0 and ω < z U.

6 6 Applied Mathematics Proof. For f MS,byTheorem 3., we know that 3. holds true. It follows that zf [ 2 Re ] ω ω z U, 3.6 where ω is analytic in U with ω 0 0and ω < z U. We now find from 3.6 that f [ 2 Re z ] ω z ω z U, 3.7 which, upon integration, yields log z [ 2 Re ] z 0 ω t dt z U. 3.8 t ω t The assertion 3.5 of Theorem 3.2 can be easily derived from 3.8. Theorem 3.3. Let f MS.Then e iθ z 2 Re / e iθ z z z 2 / 0 z U ; 0 <θ<2π. 3.9 Proof. Assume that f MS.ByTheorem 3., we know that 3. holds, which implies that zf [ / 2 Re ] e iθ z U ;0<θ<2π. 3.0 e iθ It is easy to see that the condition 3.0 can be written as follows: e iθ [ zf e iθ 2 Re ]e iθ / We note that z z z zf z z z 2 z z, 2z z z Thus, by substituting 3.2 into 3., we get the desired assertion 3.9 of Theorem 3.3.

7 Applied Mathematics 7 Theorem 3.4. Let λ Re /. Iff MS,then a k k! k 2λ j j 0 k N The inequality 3.3 is sharp for the function given by 0 <<. z z Proof. Suppose that h : / zf / i Im /. 3.5 Re / We easily know that h P. If we put h h z h 2 z 2, 3.6 it is known that h k 2 k N. 3.7 From 3.5, we have zf i Im [ Re ] h. 3.8 We now set A : i Im B : Re,. 3.9 It follows from 3.8 that zf A Bh. 3.20

8 8 Applied Mathematics Combining., 3.6,and 3.20, weobtain z z a 2 2a 2 z ka k z k Bh z Bh k z k z a 0 a z a k z k. 3.2 In view of 3.2, weget a 0 Bh 0, 3.22 ka k a k B a k h a k 2 h 2 a 0 h k h k k N From 3.7 and 3.22, weobtain a 0 2 B 2λ Moreover, we deduce from 3.7 and 3.23 that a k 2 B k k a l 2λ k k a l k N Next, we define the sequence {A k } k 0 as follows: A 0 2λ, A k 2λ k 2 k A l k N In order to prove that a k A k, 3.27 we make use of the principle of mathematical induction. By noting that a 0 2λ A Therefore, assuming that a l A l l 0,, 2,...,k; k N Combining 3.25 and 3.26,weget a k 2λ k 2 k a l 2λ k 2 k A l A k. 3.30

9 Applied Mathematics 9 Hence, by the principle of mathematical induction, we have a k A k k N as desired. By means of Lemma 2. and 3.26, we know that A k k! k 2λ j j 0 k N Combining 3.3 and 3.32, we readily get the coefficient estimates asserted by Theorem 3.4. For the sharpness, we consider the function f given by 3.4. A simple calculation shows that Re zf 2 3 z Re 2 >. z 3.33 Thus, the function f belongs to the class MS. Since 0 <<, we have λ Then f becomes z z 2λ 2λ z z n n z 2λ 2λ 2λ n z n. n! n 0 n This completes the proof of Theorem 3.4. Theorem 3.5. If f Σ satisfies the inequality k k 2 a k 2 k 0 2 <, then f MS. Proof. To prove f MS,itsuffices to show that zf 2 < 2 z U, 3.37 which is equivalent to zf 2 zf < z U. 3.38

10 0 Applied Mathematics From 3.36, we know that k a k 2 k 2 a k > k 0 k 0 Now, by the maximum modulus principle, we deduce from. and 3.39 that zf 2 zf 2 k 0 k 2 a kz k k 0 ka kz k 2 k 0 k 2 a k z k k 0 k a k z k < 2 k 0 k 2 a k k 0 k a k, 3.40 which implies that the assertion of Theorem 3.5 holds. Theorem 3.6. If f Σ satisfies the condition zf f zf < 2 2 <<, 3.4 then f MS. Proof. Define the function ϕ by ϕ : zf / zf / 2 z U Then we see that ϕ is analytic in U with ϕ 0 0. It follows from 3.42 that zf 2 ϕ ϕ By differentiating both sides of 3.43 logarithmically, we obtain zf f zf 2 zϕ [ 2 ϕ ] ϕ. 3.44

11 Applied Mathematics From 3.4 and 3.44, wefindthat zf f zf 2 zϕ [ ] 2 ϕ ϕ < Next, we claim that ϕ <. Indeed, if not, there exists a point z 0 U such that max ϕ ϕ z 0. z z By Lemma 2.2, we have ϕ z 0 e iθ, z 0 ϕ z 0 te iθ t Moreover, for z z 0,wefindfrom 3.44 and 3.47 that z 0f z 0 f z 0f z 0 z 0 f z 0 2 te iθ 2 e iθ e iθ 2 t 2 2 cos θ cos θ 2 2 <<. But 3.48 contradicts to Therefore, we conclude that ϕ <, that is zf / zf / 2 <, which shows that f MS. Acknowledgments The present investigation was supported by the National Natural Science Foundation under Grants 0053 and , the Key Project of Chinese Ministry of Education under Grant 28, the Excellent Youth Foundation of Educational Committee of Hunan Province under Grant 0B002, the Open Fund Project of the Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grants FEFM02 and 2FEFM02, and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant 2A0002 of the People s Republic of China. The authors would like to thank the referees for their careful reading and valuable suggestions which essentially improved the quality of this paper.

12 2 Applied Mathematics References M. Elin, Covering and distortion theorems for spirallike functions with respect to a boundary point, International Pure and Applied Mathematics, vol. 28, no. 3, pp , M. Elin and D. Shoikhet, Angle distortion theorems for starlike and spirallike functions with respect to a boundary point, International Mathematics and Mathematical Sciences, vol. 2006, Article ID 865, 3 pages, K. Hamai, T. Hayami, K. Kuroki, and S. Owa, Coefficient estimates of functions in the class concerning with spirallike functions, Applied Mathematics E-Notes, vol., pp , Y. C. Kim and T. Sugawa, The Alexander transform of a spirallike function, Mathematical Analysis and Applications, vol. 325, no., pp , Y. C. Kim and H. M. Srivastava, Some subordination properties for spirallike functions, Applied Mathematics and Computation, vol. 203, no. 2, pp , A. Lecko, The class of functions spirallike with respect to a boundary point, International Mathematics and Mathematical Sciences, no , pp , H. M. Srivastava, Q.-H. Xu, and G.-P. Wu, Coefficient estimates for certain subclasses of spiral-like functions of complex order, Applied Mathematics Letters, vol. 23, no. 7, pp , Y. Sun, W.-P. Kuang, and Z.-G. Wang, On meromorphic starlike functions of reciprocal order, Bulletin of the Malaysian Mathematical Sciences Society, vol. 35, no. 2, pp , N. Uyanik, H. Shiraishi, S. Owa, and Y. Polatoglu, Reciprocal classes of p-valently spirallike and p-valently Robertson functions, Inequalities and Applications, vol. 20, article 6, I. S. Jack, Functions starlike and convex of order, the London Mathematical Society, vol. 3, pp , 97.

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