STARLIKENESS ASSOCIATED WITH PARABOLIC REGIONS

STARLIKENESS ASSOCIATED WITH PARABOLIC REGIONS ROSIHAN M. ALI Received 29 April 2004 and in revised form 15 September 2004 A parabolic starlike function f of order ρ in the unit disk is characterized by the fact that the quantity zf (z)/f(z) lies in a given parabolic region in the right half-plane. Denote the class of such functions by PS (ρ). This class is contained in the larger class of starlike functions of order ρ. Subordination results for PS (ρ) are established, which yield sharp growth, covering, and distortion theorems. Sharp bounds for the first four coefficients are also obtained.there exist different extremal functions for these coefficient problems. Additionally, we obtain a sharp estimate for the Fekete-Szegö coefficient functional and investigate convolution properties for PS (ρ). 1. Introduction Let A denote the class of analytic functions f in the open unit disk U ={z : z < 1} and let f be normalized so that f (0) = f (0) 1 = 0. In [4], Goodman introduced the class UCV of uniformly convex functions consisting of convex functions f A with the property that for every circular arc γ containedin U,withcenteralsoinU, the image arc f (γ) is a convex arc. He derived a two-variable characterization of functions in UCV, that is, f A belongs to UCV if and only if for every pair (z,σ) U U, { 1+R (z σ) f } (z) 0. (1.1) f (z) Ma and Minda [6] and Rønning [10] independently developed a one-variable characterization that f UCV if and only if for every z U, ( ) zf (z) f (z) < R 1+ zf (z). (1.2) f (z) Rønning [10] also showed that f UCV if and only if the function zf PS,where PS is the class of functions g A satisfying zg (z) g(z) 1 < R zg (z), z U. (1.3) g(z) Copyright 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:4 (2005) 561 570 DOI: 10.1155/IJMMS.2005.561

562 Starlikeness associated with parabolic regions Several authors have studied the classes above, amongst which the authors of [4, 6, 7, 8, 9, 10, 12]. In [9], the class PS was generalized by looking at functions f A satisfying zf (z) f (z) 1 < R zf (z) α, z U. (1.4) f (z) In this paper, we continue the investigation of this generalized class but under a slight modification of parameter. For 0 ρ<1, let Ω ρ be the parabolic region in the right halfplane Ω ρ = { w = u + iv : v 2 < 4(1 ρ)(u ρ) } = { w : w 1 < 1 2ρ + Rw }. (1.5) The class of parabolic starlike functions of order ρ is the subclass PS (ρ)ofa consisting of functions f such that zf (z)/f(z) Ω ρ, z U.Thus f PS (ρ)ifandonlyifforz U, zf (z) f (z) 1 < 1 2ρ + R zf (z) f (z). (1.6) Similarly, a function f A belongs to UCV(ρ) if and only if for every pair (z,σ) inthe polydisk U U, { 1+R (z σ) f (z) f (z) } > 2ρ 1. (1.7) A function f UCV(ρ)iscalledanuniformly convex function of order ρ. Thus the classes discussed earlier correspond to UCV = UCV(1/2) and PS =PS (1/2). In [5], Lee showed that g UCV(ρ) f = zg PS (ρ), (1.8) that is, g UCV(ρ) zg (z) g (z) < 2(1 ρ)+r zg (z) g (z). (1.9) In the present paper, we continue the study of PS (ρ) realized by Ali and Singh [3], and more recently by Aghalary and Kulkarni [1]. We give examples of functions in the class PS (ρ), and establish subordination results, which yield sharp growth, covering and distortion theorems. Sharp bounds on the first four coefficientsare also obtained.there exist different extremal functions for these coefficient problems. Additionally, we obtain a sharp estimate for the Fekete-Szegö coefficient functional and examine convolution properties for PS (ρ). 2. Preliminary results From its definition, it is clear that the class PS (ρ) is contained in the class S (ρ) of starlike functions of order ρ, that is, R(zf (z)/f(z)) >ρ, z U. It is also fairly immediate

Rosihan M. Ali 563 that PS (ρ) is related to the class of strongly starlike functions, where a function f A is said to be strongly starlike of order α,0<α 1, if f satisfies Argzf (z)/f(z) <πα/2, z U. We state the relation in the theorem below. Theorem 2.1. If f PS (ρ), then f is strongly starlike of order γ, where(π/2)γ = tan 1 (1 ρ)/ρ. In other words, for z U, Arg zf (z) f (z) πγ 2. (2.1) Asufficient condition for a function f to be parabolic starlike of order ρ is given by the following theorem. Theorem 2.2. If f A satisfies then f PS (ρ). Proof. The given condition implies that zf (z) f (z) 1 < 1 ρ, (2.2) R zf (z) f (z) zf (z) f (z) 1 +12ρ 2(1 ρ) 2 zf (z) f (z) 1 > 0. (2.3) The following two examples are now easily established from Theorem 2.2. Example 2.3. The function f (z) = z + αz n PS (ρ)ifandonlyif α (1 ρ)/(n ρ). Example 2.4. The generalized hypergeometric function is defined by F ( a 1,...,a p ;b 1,...,b q ;z ) = 1+ n=1 ( a1 )n (a ) p n ( b1 )n (b z ) n q n n!, b j 0,1,..., (2.4) where (λ) n is the Pochhammer symbol defined by 1, n = 0, (λ) n = λ(λ +1)(λ +2) (λ + n 1), n = 1,2,... (2.5) If zf (z)/f(z) < 1 ρ,thenzf PS (ρ). Ali and Singh [3] showed that the normalized Riemann mapping function q ρ from U onto Ω ρ is given by q ρ (z) = 1+ [ 4(1 ρ) π 2 log 1+ ] z 2 1 = 1+ z B n z n. (2.6) n=1

564 Starlikeness associated with parabolic regions Here B 1 = π 2, B n = n1 nπ 2 k=0 1, n = 2,3,... (2.7) 2k +1 Since the latter sum is bounded above by 1 + (1/2)log(2n 1) (see [6]) an upper bound for each coefficient is given by B n < (1+ 1 ) nπ 2 2 log(2n 1). (2.8) However these bounds do not yield sharp coefficient estimates for the class PS (ρ). We will return to the coefficient problem in the next section. Let k PS (ρ)bedefinedbyk(0) = k (0) 1 = 0and zk (z) k(z) = q ρ(z). (2.9) In [8], Ma and Minda established a general result that leads to the following result. Theorem 2.5 [8]. If f PS (ρ), then (a) zf (z)/f(z) zk (z)/k(z) and f (z)/z k(z)/z, (b) k(r) f (z) k(r), z r <1, (c) Arg( f (z)/z) max z =r Arg(k(z)/z), z r<1, (d) k (r) f (z) k (r), z r<1. Equality in (b), (c), and (d) holds for some z 0 if and only if f is a rotation of k. Since the function k is continuous in U, k(1) = lim r 1 k(r) and k(1) = lim r 1 k(r) exist. Rønning [9] established the following corollary. Corollary 2.6 [9]. (a) Let f PS (ρ). Then either f is a rotation of k or f (U) {w : w k(1)}, where the Koebe constant is k(1) = e (1ρ)(1.25475). (b) The functions in PS (ρ) are uniformly bounded by the sharp constant k(1) = e 3.41023(1ρ). 3. Coefficient bounds We first give another sufficient condition for a function f to belong to PS (ρ). Theorem 3.1. If f (z) = z + n=2 a n z n satisfies n=2 (n 1) a n (1 ρ)/(2 ρ), then f PS (ρ). The constant (1 ρ)/(2 ρ) cannot be replaced by a larger number. Proof. Let g(z) = z 0 ( f (ξ)/ξ)dξ = z + n=2 (a n /n)z n.inviewof(1.8), it suffices to show that g UCV(ρ). Since a n 1 ρ 2 ρ, (3.1) n=2

Rosihan M. Ali 565 it follows that 1+R(zσ) g (z) n=2 g (z) 1 (n 1) a n z n2 1 n=2 a n z z σ 2ρ1. (3.2) n1 Thus g UCV(ρ). The function f (z) = z + ((1 ρ)/(2 ρ))z 2 in Example 2.3 shows that the constant (1 ρ)/(2 ρ) is the best possible. We next consider the problem of finding A n = max f PS (ρ) a n. (3.3) If f (z) = z + a 2 z 2 + a 3 z 3 + PS (ρ) andh(z) = zf (z)/f(z), then there exists a Schwarz function w defined in U with w(0) = 0, w(z) < 1, and satisfying h(z) = zf (z) f (z) = q ( ) ρ w(z). (3.4) If h(z) = 1+b 1 z + b 2 z 2 +, the first equality in (3.4) implies that n1 (n 1)a n = a k b nk. (3.5) k=1 Since q ρ is univalent in U and h q ρ, the function ( ) p(z) = 1+q1 ρ h(z) 1 q 1 ( ) = 1+c 1 z + c 2 z 2 + (3.6) h(z) ρ belongs to the class P consisting of analytic functions p in the unit disk U with positive real part such that p(0) = 1andRp(z) > 0, z U. In other words, ( ) p(z) 1 h(z) = q ρ. (3.7) p(z)+1 While (3.5)givesa n in terms of the coefficients b k,(3.7) expresses the b k s in terms of the coefficients c m s and B m s. It is now easily established that a 2 = π 2 c 1, a 3 = 2π 2 a 4 = 3π 2 [ ( 1 c 2 6 [ ( 1 c 3 3 ) π 2 12(1 ρ) π 2 c 2 1 ], ) c 1 c 2 + ( 2 45 2(1 ρ) π 2 + ) ] 32(1 ρ)2 π 4 c1 3. (3.8) Thus the coefficient estimates for PS (ρ) may be viewed in terms of nonlinear coefficient problems for the class P.

566 Starlikeness associated with parabolic regions We now introduce the following functions in PS (ρ). Define k n,g,h A, respectively, by zk n(z) k n (z) = q ( ρ z n1 ), zh ( ) (z) z(z r) H(z) = q ρ, 1 rz zg ( (z) G(z) = q ρ It is clear from (3.4)thatk n,g,h PS (ρ), and that k 2 (z) = k(z). Since we find that k n (z) = z + A n On the other hand, Ali and Singh [3]provedthat ) z(z r), 0 r 1. 1 rz (3.9) (n 1)π 2 zn +, (3.10) (n 1)π 2. (3.11) (n 1)A n 2 2(1 ρ)e 4(1ρ)2, (3.12) which also yields the sharp order of growth a n =O(1/n). From a result of Ma and Minda [8], we can also deduce the following solution to the Fekete-Szegöcoefficient functional over the class PS (ρ). We will omit the details. Theorem 3.2. Let f (z) = z + a 2 z 2 + a 3 z 3 + PS (ρ). Then [ 24(1 ρ)(1 2t)+π 2 ] 3π 4, t 1 2 π 2 96(1 ρ), a 3 ta 2 1 2 π 2, 2 π 2 96(1 ρ) t 1 2 + 5π 2 96(1 ρ), [ 24(1 ρ)(2t 1) π 2 ] 3π 4, t 1 2 + 5π 2 96(1 ρ). (3.13) If 1/2 π 2 /96(1 ρ) <t<1/2+5π 2 /96(1 ρ), equality holds if and only if f = k 3 or one of its rotations. If t<1/2 π 2 /96(1 ρ) or t>1/2+5π 2 /96(1 ρ), equality holds if and only if f = k 2 or one of its rotations. If t = 1/2 π 2 /96(1 ρ), equality holds if and only if f = H or one of its rotations, while if t = 1/2+5π 2 /96(1 ρ), then equality holds if and only if f = G or one of its rotations. The above estimates can be used to determine sharp upper bounds on the second and third coefficients, respectively, which we will state below. In addition, the sharp bound on the fourth coefficient A 4 is determined with the aid of the following lemma. Lemma 3.3 [2]. Let p(z) = 1+ k=1 c k z k P.If0 β 1 and β(2β 1) δ β, then c 3 2βc 1 c 2 + δc1 3 2. (3.14)

Rosihan M. Ali 567 In particular, c 3 2βc 1 c 2 + βc1 3 2. (3.15) When β = 0, equality holds if and only if p(z):= p 3 (z) = 3 k=1 λ k 1+ɛe 2πik/3 z 1 ɛe 2πik/3 z, ɛ =1, λ k 0, (3.16) with λ 1 + λ 2 + λ 3 = 1. Ifβ = 1, equality holds if and only if p is the reciprocal of p 3.If 0 <β<1, equality holds if and only if p(z) = 1+ɛz 1+ɛz3, ɛ =1 or p(z) =, ɛ =1. (3.17) 1 ɛz 1 ɛz3 Theorem 3.4. Let f (z) = z + a 2 z 2 + a 3 z 3 + PS (ρ). Then a 2 π 2, (3.18) with equality if and only if f = k or its rotations. Further ( 2 a 3 π 2 3 + ), π 2 0 ρ 1 π2 48, π 2, 1 π2 48 ρ<1. (3.19) For 0 ρ<1 π 2 /48, equality holds if and only if f = k or its rotations. For 1 π 2 /48 < ρ<1, equality holds if and only if f = k 3 or its rotations. If ρ = 1 π 2 /48, equality holds if and only if f = H or its rotations. Additionally, a 4 [ 12 2 3π 2 π 4 + π 2 + 23 45 3π 2, 1+ π2 16 ( ) ], 0 ρ 1+ π2 89 1, 16 45 ( ) 89 1 ρ<1. 45 (3.20) Equality holds in the upper expression of the right inequality if and only if f = k or its rotations, while equality holds in the lower expression of the right inequality if and only if f = k 4 or its rotations. Proof. In the light of Theorem 3.2, we are left to finding an estimate on the fourth coefficient. The relation (3.8)gives a 4 = 3π 2 := 3π 2 E. [ ( 1 c 3 3 ) ( 12(1 ρ) 2 2(1 ρ) π 2 c 1 c 2 + 45 π 2 + ) ] 32(1 ρ)2 π 4 c1 3 (3.21)

568 Starlikeness associated with parabolic regions We will apply Lemma 3.3 with 2β = 1 12(1 ρ) 3 π 2, δ = 2 2(1 ρ) 32(1 ρ)2 45 π 2 + π 4. (3.22) The conditions on β and δ are satisfied if ( ) 1+ π2 89 1 ρ<1. (3.23) 16 45 Thus a 4 /3π 2, with equality if and only if the function p in (3.7)isgivenby p(z) = (1 + ɛz 3 )/(1 ɛz 3 ). This implies that f = k 4. In view of the fact that 0 <δ<1, and that δ β 0provided ( ) 1+ π2 89 1 ρ, (3.24) 16 45 Lemma 3.3 yields E c 3 2δc 1 c 2 + δc1 3 +2(δ β) c 1 c 2 ( 32(1 ρ) 2 4(1 ρ) 2+8 π 4 + π 2 11 ) 90 ( 12 2 = 2 π 4 + π 2 + 23 45 ). (3.25) Equality holds if and only if the function p in (3.7) isgivenbyp(z) = (1 + ɛz)/(1 ɛz), that is, f = k. This completes the proof. Theorem 3.5. Let f (z) = z + a 2 z 2 + a 3 z 3 + PS (ρ).forµ C and λ(µ) = 1 + 3 π 2 (2µ 1), a 3 µa 2 3π 4 24(1 ρ)(1 2µ)+π 2, 2 π 2, λ(µ) 1 1, λ(µ) 1 1. (3.26) Equality holds in the upper expression of the right inequality if f = k or its rotations, while equality holds in the lower expression of the right inequality if f = k 3 or its rotations. Proof. From the relation (3.8), we get Thewell-knownestimate a 3 µa 2 2 = [ 4(1 ρ) π 2 c 2 λ(µ) ] 2 c2 1. (3.27) c 2 1 2 c2 1 2 1 2 c 1 2 (3.28)

Rosihan M. Ali 569 leads to c 2 λ(µ) 2 c2 1 c 2 1 2 c2 1 + 1 λ(µ) 2 c 1 2 λ(µ) 1 1 2+ 2 c 1 2, (3.29) which yields the desired result. 4. Convolution properties The convolution of f (z) = n=0 a n z n and g(z) = n=0 b n z n is defined to be the function ( f g)(z) = n=0 a n b n z n.forα<1, denote by R α the class of prestarlike functions of order α consisting of f A such that f ((z)/(1 z) 22α ) S (α). Here S (α) isthe class of starlike functions of order α. An important result in convolution is contained in the following lemma of Ruscheweyh. Lemma 4.1 [11, page 54]. If f R α, g S (α),andh is an analytic function in U, then f gh (U) coh(u), (4.1) f g where coh(u) is the closed convex hull of H(U). Theorem 4.2. If f R ρ and g PS (ρ), then f g PS (ρ). Proof. Since g also belongs to S (ρ)andh(z) = zg (z)/g(z) q ρ (z), Lemma 4.1 yields z( f g) (U) = f zg f g and hence, f g PS (ρ). f g (U) = f g(zg /g) f g (U) co zg g (U) Ω ρ, (4.2) Since R 1/2 = S (1/2) (see [11]), and R 0 = C, wherec is the class of convex functions in A, a similar proof also yields the following result. Corollary 4.3. (a) If f,g PS (ρ) for ρ 1/2, then f g PS (ρ). (b) If f C and g PS (ρ), then f g PS (ρ). Acknowledgments This research was supported by a Universiti Sains Malaysia Fundamental Research Grant. The author is greatly indebted to Professor V. Ravichandran for his helpful comments in the preparation of this paper. References [1] R. Aghalary and S. R. Kulkarni, Certain properties of parabolic starlike and convex functions of order ρ, Bull. Malays. Math. Sci. Soc. (2) 26 (2003), no. 2, 153 162. [2] R.M.Ali,Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc. (2) 26 (2003), no. 1, 63 71. [3] R. M. AliandV. Singh, Coefficients of parabolic starlike functions of order ρ, Computational Methods and Function Theory 1994 (Penang), Ser. Approx. Decompos., vol. 5, World Scientific Publishing, New Jersey, 1995, pp. 23 36.

570 Starlikeness associated with parabolic regions [4] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), no. 1, 87 92. [5] S. K. Lee, Characterizations of parabolic starlike functions and the generalized uniformly convex functions, Master s thesis, Universiti Sains Malaysia, Penang, Malaysia, 2000. [6] W. C. Ma and D. Minda, Uniformly convex functions, Ann. Polon. Math. 57 (1992), no. 2, 165 175. [7], Uniformly convex functions. II, Ann. Polon. Math. 58 (1993), no. 3, 275 285. [8], A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Conf. Proc. Lecture Notes Anal., I, International Press, Massachusetts, 1994, pp. 157 169. [9] F. Rønning, On starlike functions associated with parabolic regions, Ann. Univ. Mariae Curie- Skłodowska Sect. A 45 (1991), 117 122. [10], Uniformly convex functions and a corresponding class of starlike functions, Proc.Amer. Math. Soc. 118 (1993), no. 1, 189 196. [11] S. Ruscheweyh, Convolutions in Geometric Function Theory, Séminaire de Mathématiques Supérieures, vol. 83, Presses de l Université de Montréal, Quebec, 1982. [12] T. N. Shanmugam and V. Ravichandran, Certain properties of uniformly convex functions,computational Methods and Function Theory 1994 (Penang), Ser. Approx. Decompos., vol. 5, World Scientific Publishing, New Jersey, 1995, pp. 319 324. Rosihan M. Ali: School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia E-mail address: rosihan@cs.usm.my

Mathematical Problems in Engineering Special Issue on Time-Dependent Billiards Call for Papers This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome. To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome. Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www.hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable: Guest Editors Edson Denis Leonel, Department of Statistics, Applied Mathematics and Computing, Institute of Geosciences and Exact Sciences, State University of São Paulo at Rio Claro, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil; edleonel@rc.unesp.br Alexander Loskutov, Physics Faculty, Moscow State University, Vorob evy Gory, Moscow 119992, Russia; loskutov@chaos.phys.msu.ru Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009 Publication Date September 1, 2009 Hindawi Publishing Corporation http://www.hindawi.com