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Sharp Bounds on the Higher Order Schwarzian Derivatives for Janowski Classes

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Sharp Bounds on the Higher Order Schwarzian Derivatives for Janowski Classes Article Sharp Bounds on the Higher Order Schwarzian Derivatives for Janowski Classes Nak Eun Cho 1, Virendra Kumar 2,* and V. Ravichandran 3 1 Department of Applied Mathematics, Pukyong National University, Busan 48513, Korea; [email protected] 2 Department of Mathematics, Ramanujan college, University of Delhi, H-Block, Kalkaji, New Delhi 110019, India 3 Department of Mathematics, National Institute of Technology, Tiruchirappalli, Tamil Nadu 620015, India; [email protected]; [email protected] * Correspondence: [email protected] Received: 25 July 2018; Accepted: 14 August 2018; Published: 18 August 2018 Abstract: Higher order Schwarzian derivatives for normalized univalent functions were first considered by Schippers, and those of convex functions were considered by Dorff and Szynal. In the present investigation, higher order Schwarzian derivatives for the Janowski star-like and convex functions are considered, and sharp bounds for the first three consecutive derivatives are investigated. The results obtained in this paper generalize several existing results in this direction. Keywords: higher order Schwarzian derivatives; Janowski star-like function; Janowski convex function; bound on derivatives MSC: 30C45; 30C50; 30C80 1. Introduction Let A be the class of analytic functions defined on the unit disk D := fz 2 C : jzj < 1g and having the form: 2 3 f (z) = z + a2z + a3z + ··· . (1) The subclass of A consisting of univalent functions is denoted by S. An analytic function f is subordinate to another analytic function g if there is an analytic function w with jw(z)j ≤ jzj and w(0) = 0 such that f (z) = g(w(z)), and we write f ≺ g. If g is univalent, then f ≺ g if and only if ∗ f (0) = g(0) and f (D) ⊆ g(D). The classes S and K of star-like and convex functions, respectively, are among the most studied subclasses of S. These classes are defined, respectively, as: 0 ∗ z f (z) S := f 2 S : Re > 0, z 2 D f (z) and: z f 00(z) K := f 2 S : Re 1 + > 0, z 2 D . f 0(z) The Koebe function k(z) = z/(1 − z)2 2 S∗ and z/(1 − z) 2 K. General forms of these classes were considered by Janowski [1]. For −1 ≤ B < A ≤ 1, these classes are defined by: z f 0(z) 1 + Az z f 00(z) 1 + Az S∗[A, B] := f 2 S : ≺ and K[A, B] := f 2 S : 1 + ≺ . f (z) 1 + Bz f 0(z) 1 + Bz Symmetry 2018, 10, 348; doi:10.3390/sym10080348 www.mdpi.com/journal/symmetry Symmetry 2018, 10, 348 2 of 13 These classes are called the class of Janowski star-like and Janowski convex functions, respectively. On specializing the parameters A and B, we get several well-known classes such as S∗ := S∗[1, −1] and K := K[1, −1]. The functions h0 and k0 defined by: ( A − z(1 + Bz) B 1, B 6= 0; h0(z) = (2) zeAz, B = 0, and: 8 1 A [( + ) B − ] 6= 6= <> A 1 Bz 1 , B 0, A 0; k (z) = 1 0 B log(1 + Bz), A = 0; (3) :> 1 Az A [e − 1], B = 0. belong to the classes S∗[A, B] and K[A, B](−1 ≤ B < A ≤ 1), respectively. In particular, 2z/(2 − z) 2 S∗[1/2, −1/2] and 4z/(2 − z)2 2 K[1/2, −1/2]. 2 The quantity a2 − a3 is associated with the Schwarzian derivative of function f 2 S. Recall that the Schwarzian derivative of a locally univalent function f is defined by: 0 f 00(z) 1 f 00(z) 2 S( f )(z) := − f 0(z) 2 f 0(z) 2 which is an important quantity in univalent function theory. For example, the quantity a3 − ma2 = ( f 000(0) − 3m( f 00(0))2/2)/6 is called the Fekete–Szegö functional, and finding the sharp bound on modulus of this quantity is popularly known as the Fekete–Szegö problem. Nehari [2] (see also [3]) proved that the necessary condition for an analytic function f to be in the class S is jS( f )(z)j ≤ 6(1 − jzj2)−2, and the sufficient condition is jS( f )(z)j ≤ 2(1 − jzj2)−2. In both directions, the results are the best possible in the sense that the constants two and six cannot be replaced by the smaller numbers. The sharpness of the later condition was verified by Hille [4]. The first inequality is sharp in the case of the Koebe function, whereas the sharpness in the second can be seen in the case of the function f0(z) = (1/2) log((1 + z)/(1 − z)). Later, Nehari [5] proved that if f is a convex function, then jS( f )(z)j ≤ 2(1 − jzj2)−2. Aharanov and Harmelin [6] studied the higher order Schwarzian derivatives sn( f ) with invariance under composition on the left by Möbius transformations T, sn(T ◦ f ) = sn( f ), and their relation to univalence of the function f . The higher order Schwarzian derivative is defined as follows (see [6,7]): s3( f ) = S( f ) and for any integer n ≥ 4, it is given by: 00 0 f s + ( f ) = (s ( f )) − (n − 1)s ( f ) . n 1 n n f 0 In particular, f (4) f 000 f 0 f 00 3 s ( f ) = − 6 + 4 f 0 f 02 f 0 and for: f (5) f (4) f 00 f 000 2 f 000 f 02 f 00 4 s ( f ) = − 10 − 6 + 48 − 36 . 5 f 0 f 02 f 0 f 03 f 0 Schippers [7] derived the differential equation for the Loewner flow of the Schwarzian derivative of univalent functions and used this to investigate the bounds on the modulus of higher order Schwarzian derivatives. These bounds were shown to be sharp in the case of the Koebe function. He also proved certain two-point distortion theorems for the higher order Schwarzian derivatives in terms of the hyperbolic metric. Later, the higher order Schwarzian derivatives for convex functions Symmetry 2018, 10, 348 3 of 13 were considered by Dorff and Szynal [8]. Since the class K is linearly invariant (see [9]), so there is no loss in restricting consideration to sn( f )(0) =: Sn. From the above definition of sn( f ), we 2 3 see that S3 = s3( f )(0) = 6(a3 − a2), S4 = s4( f )(0) = 24(a4 − 3a2a3 + 2a2) and S5 = s5( f )(0) = 2 2 4 24(5a5 − 20a2a4 − 9a3 + 48a3a2 − 24a2). Droff and Szynal proved that jS3j ≤ 2, jS4j ≤ 4 and jS5j ≤ 12 with inequality in the case of the function: Z z n−1 − 2 fn(z) = (1 − t ) n−1 dt, n = 3, 4, 5. 0 They also conjectured that the maximal value of jSnj for n = 6, 7, 8, ··· is attained in the case of the function fn defined above. In general, it is not so easy for researchers to deal with the higher order Schwarzian derivatives as the methods in geometric function theory known at present time are not substantial enough. However, they have a very important role in geometric/univalent function theory. In particular, Gal [10], by using the powerful method of admissible functions of Miller and Mocanu [11], investigated the geometric criterion of univalence, which combines higher order Schwarzian derivatives with those of the Ruscheweyh and S˘al˘ageanoperators. Therefore, it is very natural to consider higher order Schwarzian derivatives for various geometric results of analytic functions. In this direction, Tamanoi [12], investigated various properties of higher Schwarzian derivatives and their relation with combinatorial polynomials. Tamanoi also proved that the higher Schwarzian derivatives are Möbius invariant; see [12] (p. 135 Theorems 3–3(ii)). Later, in 2011, Kim and Sugawa [13] investigated relations between the Aharonov invariants ([13,14]) and Tamanoi’s Schwarzian derivatives of higher order and gave a recursive formula for Tamanoi’s Schwarzians. In the same paper, they proposed a new definition of invariant Schwarzian derivatives of a non-constant holomorphic function between Riemann surfaces with conformal metrics. In 2011, Kim and Sugawa reviewed the Peschl–Minda derivatives [15,16] and Schwarzian derivatives of higher order due to Aharonov [14], Tamanoi [12] and Kim and Sugawa [13] for a non-constant holomorphic map between Riemann surfaces with conformal metrics. They also proved that the higher-order Schwarzian derivatives of Aharonov and Tamanoi cannot be extended to holomorphic functions between projective Riemann surfaces unlike the classical Schwarzian derivatives. The higher order Schwarzian derivative are useful in the study of the properties of non-linear dynamical system and has been studied extensively by several researchers [17]. For many applications of the higher order Schwarzian derivatives related to the real functions, the reader may refer to [17,18] and the references cited therein. Kwon and Sim [19], in 2017, using the theory of admissible functions investigated some sufficient conditions for normalized analytic functions to be star-like, associated with Tamanoi’s Schwarzian derivative of third order. Motivated by the works of Schippers [7] and Dorff and Szynal [8] and other related works cited above, in this paper, we shall consider the higher order Schwarzian derivatives for Janowski star-like and convex functions. The sharp bound on the first three consecutive Schwarzian derivatives for Janowski star-like and convex functions is investigated. We shall also point out some relevant connections of our results with the existing result. Several examples in support of our main results are also given with explanations. To prove our results, we need the following results: Let B be the class of Schwarz functions consisting of analytic functions of the form w(z) = 2 3 c1z + c2z + c3z + ··· (z 2 D) and satisfying the condition jw(z)j < 1 for z 2 D.
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