Elementary considerations for classes of meromorphic univalent functions
aa r X i v : . [ m a t h . C V ] A p r ELEMENTARY CONSIDERATIONS FOR CLASSES OFMEROMORPHIC UNIVALENT FUNCTIONS
SAMINATHAN PONNUSAMY AND KARL-JOACHIM WIRTHS
Abstract.
In this article we consider functions f meromorphic in the unit disk.We give an elementary proof for a condition that is sufficient for the univalenceof such functions. This condition simplifies and generalizes known conditions. Wepresent some typical problems of geometrical function theory and give elementarysolutions in the case of the above functions. Introduction and Main Results
Let D = { z : | z | < } and f be meromorphic in D . Much research in the lastcentury has been done concerning such functions that are injective or univalent.Many conditions that are sufficient for univalence of analytical or geometrical char-acter have been found and considered in detail. The present paper is devoted to thefollowing condition of such type that has been found by Aksent´ev in 1958, see [1],and proved again with another method by Nunokawa and Ozaki in 1972, see [5]. Theorem A.
Let f be holomorphic in D and such that f (0) = f ′ (0) − . If forall z ∈ D the inequality (1) (cid:12)(cid:12)(cid:12)(cid:12) zf ( z ) − z (cid:18) zf ( z ) (cid:19) ′ − (cid:12)(cid:12)(cid:12)(cid:12) < is valid, then f is univalent in D . The present paper is devoted to the following simple generalization of this theoremfor which we will provide an elementary proof.
Theorem 1.
Let f be meromorphic in D and such that f (0) = f ′ (0) − . If forall z ∈ D the inequality (1) is valid, then f is univalent in D . Proof.
The condition (1) implies immediately that f has no zero in D \ { } . Further,there exists a function Ω , holomorphic in D and satisfying the condition | Ω( z ) | < z ∈ D such that(2) zf ( z ) − z (cid:18) zf ( z ) (cid:19) ′ − z ) . Mathematics Subject Classification.
Key words and phrases.
Meromorphic univalent functions, coefficient estimates .File: PoWirths1˙2017˙elem.tex, printed: 14-11-2018, 5.51.The first author is currently at the ISI, Chennai Centre.
The normalization of f implies that Ω has a zero of multiplicity at least two at theorigin. Therefore we may write Ω( z ) = z ω ( z ), where ω is analytic in D such that | ω ( z ) | ≤ z ∈ D . The integration of the differential equation (2) delivers therepresentation(3) zf ( z ) = 1 + cz − z Z z ω ( t ) dt, where c is an arbitrary constant. This representation is known since long from theconsiderations of Theorem A, see for example the references in [6]. It is clear fromthe above that f satisfies (1) if and only if it has the representation (3). The proofof the univalence is a one line proof now. Consider z , z ∈ D , z = z and f ( z ) − f ( z ) z − z = 1 − z z z − z R z z ω ( t ) dt (1 + cz − z R z ω ( t ) dt )(1 + cz − z R z ω ( t ) dt ) . Since(4) (cid:12)(cid:12)(cid:12)(cid:12)Z z z ω ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ | z − z | , we see that f ( z ) − f ( z ) = 0. This proves the univalence of f . (cid:3) Remark 1.
The difference between Theorem A and Theorem 1 is the fact that inTheorem 1, the function f is allowed to have a pole in the unit disk. The proof ofTheorem 1 shows that such pole is necessarily a simple pole.We want to add the hint that similar meromorphic functions have been consideredin [2].In what follows, we shall derive some analytic properties of the meromorphicfunctions satisfying (1). To that end we refine this condition, namely, we let λ ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12) zf ( z ) − z (cid:18) zf ( z ) (cid:19) ′ − (cid:12)(cid:12)(cid:12)(cid:12) < λ, z ∈ D . For more details on the holomorphic functions satisfying such inequalities, we referagain to [6] and the references therein.
Theorem 2. (a)
Let f be meromorphic in D and such that f (0) = f ′ (0) − .If for all z ∈ D the inequality (1) is valid, then f is continuous in D with apossible exception at a zero of (3) . (b) Let f be meromorphic in D and such that f (0) = f ′ (0) − . If for all z ∈ D the inequality (5) is valid for some λ ∈ (0 , , then f is univalent in D . Proof.
For the proof of (a) it is sufficient to recognize that (4) implies that A ( z ) = 1 + cz − z Z z ω ( t ) dt eromorphic univalent functions 3 is uniformly continuous in D . This implies the continuity of this function in D andthis in turn implies the continuity of f in D with the exception of the zeros of A ( z ).In the proof of (b) we use the fact that A λ ( z ) = 1 + cz − λz Z z ω ( t ) dt has a continuous extension to D for λ ∈ [0 ,
1] which may be denoted by A λ ( z ), too.For z , z ∈ D , z = z , we get as in the proof of Theorem 1 that (cid:12)(cid:12)(cid:12)(cid:12) z A λ ( z ) − z A λ ( z ) z − z (cid:12)(cid:12)(cid:12)(cid:12) ≥ − λ | z z | > λ ∈ [0 , f ( z ) − f ( z ) z − z = 0 , and therefore the assertion of the theorem is proved. (cid:3) Another typical problem in geometric function theory is the estimation of themoduli of the Taylor coefficients. Since in our case, the functions f are holomorphicin a neighbourhood of the origin, we may ask for the upper bounds of the coefficientsof the Taylor expansion in a disk around the origin. For the second coefficient wecan give a satisfactory result using an elementary method from [6]. Also, comparewith [2], too. Theorem 3.
Let λ ∈ (0 , and f be meromorphic in D and such that f (0) = f ′ (0) − . If for all z ∈ D the inequality (5) is valid and f ( z ) = z + ∞ X n =2 a n z n has no pole in the disk { z : | z | < p } , p ∈ (0 , , then the inequality (6) | a | ≤ λp p is valid. The inequality is sharp and equality is attained if and only if (7) f ( z ) = z (cid:16) − e iθ zp (cid:17) (1 − λpe iθ z ) , θ ∈ [0 , π ) . Proof.
From the hypotheses above, we know that f is of the form f ( z ) = z cz − λz R z ω ( t ) dt . Obviously, a = − c . Now, we assume on the contrary that | a | > λp p . Then we can define | a | = 1 + λp pr S. Ponnusamy and K.-J. Wirths for some r ∈ (0 , F ( z ) = 1 a (cid:18) − λz Z z ω ( t ) dt (cid:19) in the closed disk D rp := { z : | z | ≤ pr } . Then we have the inequality | F ( z ) | ≤ pr (1 + λp r )1 + λp < pr which implies that F maps this disk into itself. Secondly, for z , z in the disk D rp ,we have | F ( z ) − F ( z ) | = λrp λp (cid:12)(cid:12)(cid:12)(cid:12) z Z z ω ( t ) dt − ( z − z + z ) Z z ω ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ λrp λp (cid:18) | z | (cid:12)(cid:12)(cid:12)(cid:12)Z z z ω ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) + | z − z | (cid:12)(cid:12)(cid:12)(cid:12)Z z ω ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ λrp ( | z | + | z | ) | z − z | λp ≤ λr p λp | z − z | . Since 2 λr p / (1 + λp ) < , the conditions of Banach’s fixed point theorem arefulfilled and thus, the function F has a fixed point in the closed disk { z : | z | ≤ pr } .Therefore f has a pole in the disk D rp . This is a contradiction to the conditionsof Theorem 3. It is easy to see that the functions (7) satisfy the conditions of thetheorem and that for them equality in (6) is attained. To prove the rest of theassertion, we use a method from [6], where this has been shown in the case p = 1.Therefore, we have to consider here only the cases p <
1. Let us assume that thereexists a function ω that is not of constant modulus one and is such that the function f ( z ) = z − λp p e iθ z − λz R z ω ( t ) dt satisfies the conditions of the theorem. According to the above assumption, thereexists a number c ∈ [0 ,
1) such that for | z | ≤ p the inequality (cid:12)(cid:12)(cid:12)(cid:12)Z z ω ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | z | is valid. We define r = 1 + λcp λp ∈ (0 , . Now, we consider again the function (8) and we get by considerations similar to theabove ones that F is a contractive mapping of the closed disk { z : | z | ≤ pr } intoitself. Hence, F has a fixed point in this disk, and therefore there exists a pole of f in the same disk. This again contradicts the above conditions. Now, it remains toprove that the only quadratic polynomials of the form1 − λp p e iθ z − λe iϕ z eromorphic univalent functions 5 that have no zero in the disk { z : | z | < p } are the polynomials1 − λp p e iθ z + λe iθ z . This is a simple exercise of calculations and thus, we complete the proof. (cid:3)
Remark 2.
In a neighbourhood of the origin, the functions (7) have the Taylorexpansion f ( z ) = ∞ X n =1 − λ n p n p n − (1 − λp ) e inθ z n . We conjecture that this delivers the upper bound | a n | ≤ − λ n p n p n − (1 − λp )for all functions f satisfying the conditions of Theorem 3. In the case λ = 1 and p = 1, this is a special case of the famous Bieberbach conjecture that has beenproved by de Branges in [3]. For λ = 1 and arbitrary p ∈ [0 ,
1) the validity ofthe Bieberbach conjecture implies that the above conjecture is true. This has beenproved by Jenkins in [4]. For λ ∈ [0 ,
1) and p = 1 the truth of the above conjecturefor n = 3 and n = 4 was shown in [7]. However the general conjecture remains open.We conclude the article by recalling that the above conjecture is a generalization ofthe following conjecture from [7]. Conjecture 1.
Suppose that a holomorphic function f in satisfies the condition (5) for some < λ ≤ . Then | a n | ≤ P n − k =0 λ k for n ≥ . References
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On a subclass of meromorphic univalent functions , Complex Var.Elliptic Equ. (2017), 494–510.3. L. de Branges, A proof of the Bieberbach conjecture , Acta Math. (1985), 137–152.4. J. A. Jenkins, On a conjecture of Goodman concerning meromorphic univalent functions , Michi-gan Math. J. (1962), 25–27.5. M. Nunokawa and S. Ozaki, The Schwarzian derivative and univalent functions , Proc. Amer.Math. Soc. (1972), 392–394.6. M. Obradovi´c, S. Ponnusamy, and K.-J. Wirths, Geometric studies on the class U ( λ ), Bull.Malaysian Math. Sci. Soc. (2016), 1259–1284.7. M. Obradovi´c, S. Ponnusamy, and K.-J. Wirths, Logarithmic coefficients and a coefficientconjecture for univalent functions , Monatshefte Math. (2017).Online DOI 10.1007/s00605-017-1024-3
S. Ponnusamy, Department of Mathematics, Indian Institute of Technology Madras,Chennai-600 036, India.
E-mail address : [email protected], [email protected] K.-J. Wirths, Institut f¨ur Analysis und Algebra, TU Braunschweig, 38106 Braun-schweig, Germany.
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