IJANAL International Journal of Analysis 2314-4998 2314-498X Hindawi Publishing Corporation 10.1155/2015/185635 185635 Research Article Geometric Properties of a Class of Analytic Functions Defined by a Differential Inequality Kaur Manpreet Gupta Sushma Singh Sukhjit Cahlon Baruch Department of Mathematics Sant Longowal Institute of Engineering and Technology Longowal Punjab 148106 India sliet.ac.in 2015 4102015 2015 08 07 2015 10 09 2015 15 09 2015 4102015 2015 Copyright © 2015 Manpreet Kaur 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.

Let A be the class of analytic functions f defined in the open unit disk E and normalized by f ( 0 ) = f ' ( 0 ) - 1 = 0 . For f ( z ) / z 0 in E , let M α , λ : = { f A : | - α z 2 ( z / f ( z ) )    + f ( z ) ( z / f ( z ) ) 2 - 1 | λ , z E } , where λ > 0 and α R [ - 1 / 2,0 ] . In the present paper, we find conditions under which functions in the class M ( α , λ ) are starlike of order γ , 0 γ < 1 .

1. Introduction

Let A denote the class of all functions f that are analytic in the open unit disk E = { z : z < 1 } and are normalized by the conditions f ( 0 ) = f ( 0 ) - 1 = 0 . We denote by S the subclass of A consisting of functions which are univalent in E and denote by S ( γ ) , 0 γ < 1 , the class of functions in S which are starlike of order γ in E . Analytically, (1) S γ = f A : Re z f z f z > γ , z E . Note that S ( 0 ) is the usual class of starlike (with respect to the origin) functions in E and we denote it simply by S . We say that f R ( β ) , 0 < β 1 , if f A and a r g f z < β π / 2 , for all z E . It is well known that the functions in R ( 1 ) are close-to-convex and hence univalent in E [1, 2].

In 1972, Ozaki and Nunokawa  studied the class U ( λ ) , which is defined as (2) U λ = f A : f z z f z 2 - 1 λ , z E , where f ( z ) / z 0 in E . They proved that U ( λ ) S for 0 λ 1 . Several researchers studied this class (e.g., see ) and obtained many significant results.

Recently, Obradovic and Ponnusamy  investigated another class (3) M λ = f A : z 2 z f z + f z z f z 2 - 1 λ , z E , where λ > 0 and f ( z ) / z 0 in E . They obtained certain inclusion relations, characterization formula, and coefficient conditions. They also posed a question about the starlikeness of the functions in the class M ( λ ) . The purpose of the present paper is to answer this question. In fact, we study a more general class (4) M α , λ = f A : - α z 2 z f z + f z z f z 2 - 1 λ , z E , where f ( z ) / z 0 in E , λ > 0 , and α R [ - 1 / 2,0 ] . We find conditions on α , λ , and f involved in the class M ( α , λ ) under which members of M ( α , λ ) are starlike of a given order γ , 0 γ < 1 . We remark that the class M ( α , λ ) follows essentially from the class S ( α , β , λ ) studied by Baricz and Ponnusamy  by taking β = 0 ; however, we will study those issues for the class M ( α , λ ) which are not studied by the authors for S ( α , β , λ ) .

2. Main Results

Let A n denote the class of analytic functions p in E such that p ( k ) ( 0 ) = 0 for k = 0,1 , 2 , , n , where p ( 0 ) ( 0 ) = p ( 0 ) . With w ( 0 ) ( 0 ) = w ( 0 ) , we set B n = { w : w is analytic, w z 1 in E and w ( k ) ( 0 ) = 0 for k = 0,1 , 2 , , n } . Functions in B n are called Schwarz functions. Obviously, w B k implies that | w ( z ) | | z | k + 1 in E , for k = 0,1 , 2,3 , , n .

We begin with the following result.

Lemma 1.

Let f ( z ) = z + a 2 z 2 + a 3 z 3 + be in M ( α , λ ) . Then, R e f ( z ) / z 1 / 2 in E whenever a 2 1 - λ / 2 α + 1 .

Proof.

As f M ( α , λ ) , there exists a Schwarz function w B 1 such that (5) - α z 2 z f z + f z z f z 2 - 1 = λ w z . Since w B 1 , w ( 0 ) = w ( 0 ) = 0 . If we set (6) p z = f z z f z 2 - 1 = z f z - z z f z - 1 , where p is an analytic function in E with p ( 0 ) = p ( 0 ) = 0 , then (5) is equivalent to (7) α z p z + p z = λ w z , from which we get (8) p z = λ α 0 1 w t z t 1 / α - 1 d t , or, equivalently, (9) - z z f z - 1 + a 2 z + z f z - 1 + a 2 z = λ α 0 1 w t z t 1 / α - 1 d t . Solving this equation for z / f ( z ) , we obtain (10) z f z = 1 - a 2 z - λ α 0 1 0 1 w s t z s 2 t 1 / α - 1 d s d t . Now, using the fact that w s t z s t z 2 , it follows that (11) z f z - 1 a 2 z + λ α 0 1 0 1 t 1 / α + 1 z 2 d s d t , z E , z a 2 + λ z 2 α + 1 , z E .

The inequality (11) is equivalent to (12) f z z - 1 1 - z 2 a 2 + λ z / 2 α + 1 2 z a 2 + λ z / 2 α + 1 1 - z 2 a 2 + λ z / 2 α + 1 2 , which gives (13) R e f z z 1 1 + z a 2 + λ z / 2 α + 1 > 1 1 + a 2 + λ / 2 α + 1 1 2 , for all z E whenever a 2 1 - λ / 2 α + 1 .

In the next result, we find the range of values of λ for which f M ( α , λ ) implies that f R ( β ) , 0 < β 1 .

Theorem 2.

Let f ( z ) = z + a 2 z 2 + a 3 z 3 + be in M ( α , λ ) . Then, f R ( β ) for 0 < λ λ β , where λ β satisfies the inequality (14) 1 - λ 2 2 α + 1 2 sin β π 2 - 2 a 2 + λ 2 α + 1 1 - a 2 + λ 2 α + 1 2 λ 2 α + 1 cos β π 2 .

Proof.

From (8), we get (15) p z < λ 2 α + 1 , z E , where (16) p z = f z z f z 2 - 1 . Therefore, (17) arg f z z f z 2 arcsin λ 2 α + 1 , z E . Also, in view of (11), we obtain (18) arg z f z arcsin a 2 + λ 2 α + 1 , z E . Therefore, (19) arg f z arg f z z f z 2 + 2 arg z f z arcsin λ 2 α + 1 + 2 arcsin a 2 + λ 2 α + 1 = arcsin λ 2 α + 1 + arcsin 2 a 2 + λ 2 α + 1 1 - a 2 + λ 2 α + 1 2 . Now, the desired result follows, if (20) a r c s i n λ 2 α + 1 + a r c s i n 2 a 2 + λ 2 α + 1 1 - a 2 + λ 2 α + 1 2 β π 2 . By using a r c s i n x + a r c s i n y = a r c s i n x 1 - y 2 + y 1 - x 2 , x , y - 1,1 , and x 2 + y 2 1 and carrying out some simplifications, we conclude that (20) is equivalent to (14).

If a 2 = 0 and β = 1 , then Theorem 2 gives the following.

Corollary 3.

If f ( z ) = z + a 3 z 3 + is in M ( α , λ ) , and then R e f ( z ) > 0 in E , whenever 0 < λ 2 α + 1 / 2 .

Setting a 2 = 0 , β = 1 , and α = - 1 in Theorem 2, we obtain the following.

Corollary 4.

If f ( z ) = z + a 3 z 3 + is in M ( - 1 , λ ) , and then R e f ( z ) > 0 in E , whenever 0 < λ 1 / 2 .

In the following theorem, we find conditions under which functions in the class M ( α , λ ) belong to S ( γ ) , 0 γ < 1 .

Theorem 5.

Let f ( z ) = z + a 2 z 2 + a 3 z 3 + be in M ( α , λ ) and let f ( 0 ) = 0 . Then, for 0 γ < 1 , f S ( γ ) provided 0 < λ 1 - γ 2 α + 1 / ( 2 + γ ) .

Proof.

As a 2 = f ( 0 ) / 2 = 0 , from (8) and (10), we get (21) z f z f z = f z z / f z 2 z / f z = 1 + λ / α 0 1 w t z t 1 / α - 1 d t 1 - λ / α 0 1 0 1 w s t z / s 2 t 1 / α - 1 d s d t . Now, R e z f ( z ) / f ( z ) > γ is equivalent to (22) z f z / f z - γ 1 - γ i T , T R . Using (21) in (22), we get (23) λ α 0 1 w t z t 1 / α - 1 d t - γ 1 - i T 1 - λ α 0 1 0 1 w s t z s 2 t 1 / α - 1 d s d t + λ α i T 0 1 0 1 w s t z s 2 t 1 / α - 1 d s d t - 1 - i T , or, equivalently, (24) λ 2 α 0 1 w t z t 1 / α - 1 d t - 0 1 0 1 w s t z s 2 t 1 / α - 1 d s d t - γ 1 - λ α 0 1 0 1 w s t z s 2 t 1 / α - 1 d s d t + λ 2 α 1 + i T 1 - i T 0 1 w t z t 1 / α - 1 d t + 0 1 0 1 w s t z s 2 t 1 / α - 1 d s d t - 1 . If we denote the left-hand side of (24) by H ( w , T , z ) and let (25) M = sup T R , w B 1 , z E H w , T , z , then, in view of the rotation invariance property of the set B 1 , we obtain that (22) holds if M 1 .

A simple calculation gives (26) M sup w B 1 , z E λ 2 α 0 1 w t z t 1 / α - 1 d t - 0 1 0 1 w s t z s 2 t 1 / α - 1 d s d t + λ 2 α 0 1 w t z t 1 / α - 1 d t + 0 1 0 1 w s t z s 2 t 1 / α - 1 d s d t + γ 1 - λ α 0 1 0 1 w s t z s 2 t 1 / α - 1 d s d t . Now, by the parallelogram Law, z 1 + z 2 + z 1 - z 2 2 z 1 2 + z 2 2 , we have (27) M λ α sup w B 1 , z E 0 1 w t z t 1 / α - 1 d t 2 + 0 1 0 1 w s t z s 2 t 1 / α - 1 d s d t 2 + γ + λ γ α 0 1 0 1 w s t z s 2 t 1 / α - 1 d s d t . Using the fact that w z z 2 in E , we get (28) M λ α sup z E 0 1 z 2 t 1 / α + 1 d t 2 + 0 1 0 1 z 2 t 1 / α + 1 d t 2 + γ + λ γ α 0 1 0 1 z 2 t 1 / α + 1 d t = λ 2 + γ 2 α + 1 + γ . Thus, M 1 , if λ 1 - γ 2 α + 1 / ( 2 + γ ) .

Taking γ = 0 , in Theorem 5, we get the following result.

Corollary 6.

Let f be as in Theorem 5. Then, f S provided 0 < λ 2 α + 1 / 2 .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Noshiro K. On the theory of Schlicht functions Journal of the Faculty of Science, Hokkaido University 1934-1935 2 3 129 155 Warchawski S. E. On the higher derivatives at the boundary in conformal mapping Transactions of the American Mathematical Society 1935 38 2 310 340 10.2307/1989685 Ozaki S. Nunokawa M. The Schwarzian derivative and univalent functions Proceedings of the American Mathematical Society 1972 33 2 392 394 10.1090/s0002-9939-1972-0299773-3 MR0299773 Obradović M. A class of univalent functions Hokkaido Mathematical Journal 1998 27 2 329 335 10.14492/hokmj/1351001289 MR1638004 Obradovic M. Ponnusamy S. Singh V. Vasundhra P. Univalency, starlikeness and convexity applied to certain classes of rational functions Analysis: International Mathematical Journal of Analysis and its Applications 2002 22 3 225 242 10.1524/anly.2002.22.3.225 MR1938375 Ponnusamy S. Vasundhra P. Univalent functions with missing Taylor coefficients Hokkaido Mathematical Journal 2004 33 2 341 355 10.14492/hokmj/1285766169 MR2073002 Ponnusamy S. Vasundhra P. Criteria for univalence, starlikeness and convexity Annales Polonici Mathematici 2005 85 2 121 133 10.4064/ap85-2-2 MR2180658 Singh V. On a class of univalent functions International Journal of Mathematics and Mathematical Sciences 2000 23 12 855 857 10.1155/s0161171200001824 MR1769871 Obradovic M. Ponnusamy S. A class of univalent functions defined by a differential inequality Kodai Mathematical Journal 2011 34 2 169 178 10.2996/kmj/1309829544 MR2811638 2-s2.0-79960106875 Baricz Á. Ponnusamy S. Differential inequalities and Bessel functions Journal of Mathematical Analysis and Applications 2013 400 2 558 567 10.1016/j.jmaa.2012.11.050 MR3004986 2-s2.0-84872016390