Mil1final.dvi CUBO A Mathematical Journal Vol.12, No¯ 01, (15–21). March 2010 On Some Problems of James Miller B. Bhowmik, S. Ponnusamy Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India emails: ditya@iitm.ac.in, samy@iitm.ac.in and K.-J. Wirths Institut für Analysis, TU Braunschweig, 38106 Braunschweig, Germany email : kjwirths@tu-bs.de ABSTRACT We consider the class of meromorphic univalent functions having a simple pole at p ∈ (0, 1) and that map the unit disc onto the exterior of a domain which is starlike with respect to a point w0 6= 0, ∞. We denote this class of functions by Σ ∗(p, w0). In this paper, we find the exact region of variability for the second Taylor coefficient for functions in Σ∗(p, w0). In view of this result we rectify some results of James Miller. RESUMEN Consideramos la clase de funciones univalentes meromoforficos teniendo un polo simple en p ∈ (0, 1) y la aplicación del disco unitario sobre el exterior de un dominio el cual es estrellado con respecto al punto w0 6= 0, ∞. Denotamos esta clase de funciones por Σ ∗(p, w0). En este art́ıculo encontramos la región exacta de variabilidad del segundo coeficiente de Taylor para funciones in Σ∗(p, w0). En vista de estos resultados nosotros rectificamos algunos resultados de James Miller. 16 B. Bhowmik, S. Ponnusamy and K.-J. Wirths CUBO 12, 1 (2010) Key words and phrases: Starlike, meromorphic, and Schwarz’ functions, Taylor coefficient. Math. Subj. Class.: 30C45. 1 Introduction Let D := {z : |z| < 1} be the unit disc in the complex plane C. Let Σ∗ denote the class of functions g(z) = 1 z + d0 + d1z + d2z 2 + · · · which are univalent and analytic in D except for the simple pole at z = 0 and map D onto a domain whose complement is starlike with respect to the origin. Functions in this class is referred to as the meromorphic starlike functions in D. This class has been studied by Clunie [4] and later an extended version by Pommerenke [10], and many others. Another related class of our interest is the class S(p) of univalent meromorphic functions f in D with a simple pole at z = p, p ∈ (0, 1), and with the normalization f (z) = z + ∑∞ n=2 an(f )z n for |z| < p. If f ∈ S(p) maps D onto a domain whose complement with respect to C is convex, then we call f a concave function with pole at p and the class of these functions is denoted by Co(p). In a recent paper, Avkhadiev and Wirths [2] established the region of variability for an(f ), n ≥ 2, f ∈ Co(p) and as a consequence two conjectures of Livingston [7] in 1994 and Avkhadiev, Pommerenke and Wirths [1] were settled. In this paper, we consider the class Σ∗(p, w0) of meromorphically starlike functions f such that C \ f (D) is a starlike set with respect to a finite point w0 6= 0 and have the standard normalization f (0) = 0 = f ′(0) − 1. We now recall the following analytic characterization for functions in Σ∗(p, w0). Theorem A. f ∈ Σ∗(p, w0) if and only if there is a probability measure µ(ζ) on ∂D = {ζ : |ζ| = 1} so that f (z) = w0 + pw0 (z − p)(1 − zp) exp ( ∫ ∂D 2 log(1 − ζz)dµ(ζ) ) where w0 = − 1 p + 1/p − 2 ∫ |ζ|=1 ζdµ(ζ) . The necessary part of Theorem A has been proved by Miller [9] while the sufficiency part has been established by Yuh Lin [6, Theorem 1]. In [8, 9], Miller discussed a numbers of properties of the class Σ∗(p, w0). See also [3, 6, 11] for some other basic results such as bounds for |f (z) − w0|. We may state an equivalent formulation of Theorem A (see also [11]). A function f is said to be in Σ∗(p, w0) if and only if there exists an analytic function P (z) in D with P (0) = 1 and Re P (z) > 0, z ∈ D, (1.1) CUBO 12, 1 (2010) On Some Problems of James Miller 17 where P (z) = −zf ′(z) f (z) − w0 − p z − p + pz 1 − pz . (1.2) We may write P (z) in the following power series form P (z) = 1 + b1z + b2z 2 + · · · . Also, each f ∈ Σ∗(p, w0) has the Taylor expansion f (z) = z + ∞ ∑ n=2 an(f )z n, |z| < p. (1.3) To recall the next result, we need to introduce a notation. Let P(b1) denote the class of analytic functions P (z) satisfying P (0) = 1, P ′(0) = b1 and Re P (z) > 0 in D. In 1972, Miller [8] obtained estimations for the second Taylor coefficient a2(f ). Indeed, he showed that Theorem B. If f (z) ∈ Σ∗(p, w0), then the second coefficient is given by a2(f ) = 1 2 w0 ( b2 − p 2 − 1 p2 − 1 w02 ) where b2 is the second coefficient of a function in P(b1), i.e. the region of variability for a2(f ) is contained in the disc ∣ ∣ ∣ ∣ a2(f ) + 1 2 w0 ( p2 + 1 p2 + 1 w02 ) ∣ ∣ ∣ ∣ ≤ |w0|. (1.4) Further there is a p0, 0.39 < p0 < 0.61, such that if p < p0, then Re a2(f ) > 0. In 1980, Miller [9, Equation (9)] also proved a sharp estimate regarding the second Taylor coefficient. In fact, he showed that ∣ ∣ ∣ ∣ a2(f ) − 1 + p2 p − w0 ∣ ∣ ∣ ∣ ≤ |w0|, f ∈ Σ ∗(p, w0). (1.5) The aim of this paper is to find the region of variability for the second coefficient a2(f ) of functions in Σ∗(p, w0) for any fixed pair (p, w0). Also we find the exact region of variability for a2(f ) for fixed p, and as a consequence of this we show that Re a2(f ) > 0 for all values of p ∈ (0, 1) which Miller did not seem to expect as we see in the last part of Theorem B. 2 Region of Variability of Second Taylor Coefficients for Functions in Σ∗(p, w0) Theorem 2.1. Let f ∈ Σ∗(p, w0) having the expansion (1.3). Then for a fixed pair (p, w0), the exact region of variability of the second Taylor coefficient a2(f ) is the disc determined by the 18 B. Bhowmik, S. Ponnusamy and K.-J. Wirths CUBO 12, 1 (2010) inequality ∣ ∣ ∣ ∣ ∣ a2(f ) − ( p + 1 p + w0 ) + 1 4 w0 ( p + 1 p + 1 w0 )2 ∣ ∣ ∣ ∣ ∣ ≤ |w0| ( 1 − 1 4 ∣ ∣ ∣ ∣ p + 1 p + 1 w0 ∣ ∣ ∣ ∣ 2 ) . (2.2) Proof. The proof uses the representation formula (1.1), i.e. f ∈ Σ∗(p, w0) if and only if Re P (z) > 0 in D with P (0) = 1, where P is given by (1.2). Since it is convenient to work with the class of Schwarz functions, we can write each such P as P (z) = 1 + ω(z) 1 − ω(z) , z ∈ D, (2.3) where ω : D → D is holomorphic with ω(0) = 0 so that ω(z) has the form ω(z) = c1z + c2z 2 + · · · . (2.4) Using (1.2) and the power series representations of P (z) and f (z), it is easy to compute      b1 = p + 1 p + 1 w0 , and b2 = p 2 + 1 p2 + 1 w02 + 2a2(f ) w0 . (2.5) Now eliminating w0 from (2.5), we get b2 = p 2 + 1 p2 + [ b1 − ( p + 1 p )]2 + 2a2(f ) [ b1 − ( p + 1 p )] . (2.6) Using the power series representations of P (z) and ω(z), it follows by comparing the coefficients of z and z2 on both sides that b1 = 2c1 and b2 = 2(c 2 1 + c2). Inserting the above two relations in (2.6), we get 2(c2 1 + c2) = p 2 + 1 p2 + [ 2c1 − ( p + 1 p )]2 + 2a2(f ) [ 2c1 − ( p + 1 p )] . Now solving the above equation for a2(f ), we get a2(f ) = 1 p + p ( c2 1 − c2 + p 2 − 2c1p 1 + p2 − 2c1p ) . (2.7) Now, since w0 and p are fixed, we have c1 fixed. Hence using the well known estimate |c2| ≤ 1−|c1| 2 for unimodular bounded function ω(z), the last equation results the following estimate ∣ ∣ ∣ ∣ a2(f ) − 1 p − p ( c2 1 + p2 − 2c1p 1 + p2 − 2c1p ) ∣ ∣ ∣ ∣ ≤ p(1 − |c1| 2) |1 + p2 − 2c1p| . CUBO 12, 1 (2010) On Some Problems of James Miller 19 Now, as b1 = 2c1, substituting c1 = 1 2 (p + 1/p + 1/w0) in the above equation we get the required estimate as given in (2.2). A point on the boundary of the disc described by (2.2) is attained for the unique extremal functions given by (1.2) and (2.3), where ω(z) = z(c1 + cz) 1 + c1cz , |c| = 1. The points in the interior of the disc described in (2.2) are attained for the same functions, but with |c| < 1. Remark. Comparison of Theorem B and Theorem 2.8 below, shows that the exact region of variability of a2(f ) found by Miller is for the case c1 = 0 only. A little computation reveals that both variability regions are the same for c1 = 0, i.e., ∣ ∣ ∣ ∣ a2(f ) − 1 + p2 + p4 p(1 + p2) ∣ ∣ ∣ ∣ ≤ p 1 + p2 . This also shows that (1.5) gives the precise region of variability only for the case c1 = 0. In all other cases, the boundaries of the discs described by (1.4) and (1.5) have only one point in common with the disc described by (2.2) because, in the both cases, on the boundaries of the discs described by (1.4) and (1.5), we need |b2| = 2. Now, as b2 = 2(c2 + c1 2), this means that |c2 + c1 2| = 1. According to the coefficients bounds for unimodular bounded function, this is only possible for a unique c2 if c1 6= 0. In the following theorem, we describe the exact region of variability of the second Taylor coefficient of f ∈ Σ∗(p, w0), where only p is fixed. Theorem 2.8. Let f ∈ Σ∗(p, w0) having the expansion (1.3) and let p be fixed. Then the exact set of variability of the second Taylor coefficient a2(f ) is given by |a2(f ) − 1/p| ≤ p. (2.9) Proof. We may rewrite (2.7) as a2(f ) = 1 p + p M, (2.10) where M = c2 1 − c2 + p 2 − 2c1p 1 + p2 − 2c1p . We wish to prove that |M| ≤ 1. Since ω′(0) = c1, we have |c1| ≤ 1. Now we fix c1 ∈ D. Then c 2 1 − c2 varies in the closed disc ∆(c1) := {z : |z − c 2 1 | ≤ 1 − |c1| 2}. 20 B. Bhowmik, S. Ponnusamy and K.-J. Wirths CUBO 12, 1 (2010) The map T (ζ) = ζ + p2 − 2c1p 1 + p2 − 2c1p maps the disc ∆(c1) onto the disc with center c2 1 + p2 − 2c1p 1 + p2 − 2c1p and radius 1 − |c1| 2 |1 + p2 − 2c1p| . Therefore, in order to prove |M| ≤ 1, it suffices to show that ∣ ∣ ∣ ∣ c2 1 + p2 − 2c1p 1 + p2 − 2c1p ∣ ∣ ∣ ∣ + 1 − |c1| 2 |1 + p2 − 2c1p| ≤ 1. This is equivalent to |c1 − p| 2 + 1 − |c1| 2 = Re (1 + p2 − 2c1p) ≤ |1 + p 2 − 2c1p|. We see that equality is attained in the above inequality if and only if c1 is real. Now for real c1, we have T (∆(c1)) = D if and only if c1 = p or w0 = −p 1 − p2 . Hence the extremal functions for the inequality (2.9) are given by (1.2) with P (z) as in (2.3) with ω(z) = z(p + cz) 1 + pcz , |c| = 1, and the points in the interior of the disc described by (2.9) are attained for the same functions, but with |c| < 1. We observe that for real c1 we can obtain M = 1 only for c2 = c 2 1 − 1. This results in other starlike centers, but the extremal function is always the same, since a2(f ) = p + 1/p is attained in the class S(p) only for f (z) = z/((1 − zp)(1 − z/p)), see for instance [5]. Remark. This result ensures us that Re a2(f ) > 0 for all p ∈ (0, 1). In the article [8, Theorem 1], Miller hoped for a possibility that for p > .61, the real part of a2(f ) may be negative. But in view of our theorem we conclude that his hope was in vain. Remark. In [9], Miller has obtained an estimate for the real part of the third coefficient a3(f ) for all p. However, in geometric function theory, the classical question of finding the exact region of variability for an(f ), n ≥ 3, f ∈ Σ ∗(p, w0), remains an open problem. Received: March, 2008. Revised: September, 2009. CUBO 12, 1 (2010) On Some Problems of James Miller 21 References [1] Avkhadiev, F.G., Pommerenke, C. and Wirths, K.-J., On the coefficients of concave univalent functions, Math. Nachr., 271(2004), 3–9. [2] Avkhadiev, F.G. and Wirths, K.-J., A proof of the Livingston conjecture, Forum math., 19(2007), 149–158. [3] Bhowmik, B. and Ponnusamy, S., Coefficient inequalities for concave and meromorphically starlike univalent functions, Ann. Polon. Math., 93(2008), 177–186. [4] Clunie, J., On meromorphic schlicht functions, J. London Math. Soc., 34(1959), 215–216. [5] Jenkins, J.A., On a conjecture of Goodman concerning meromorphic univalent functions, Michigan Math. J., 9(1962), 25–27. [6] Yuh Lin, C., On the representation formulas for the functions in the class Σ∗(p, w0), Proc. 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