HARMONIC CLOSE-TO-CONVEX MAPPINGS

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1 Applied Mathematics Stochastic Analysis, 15:1 (2002, HARMONIC CLOSE-TO-CONVEX MAPPINGS JAY M. JAHANGIRI 1 Kent State University Department of Mathematics Burton, OH USA jay@geauga.kent.edu HERB SILVERMAN College of Charleston Department of Mathematics Charleston, SC USA silvermanh@cofc.edu (Received October, 1999; Revised May, 2000 Sufficient coefficient conditions for complex functions to be close-to-convex harmonic or convex harmonic are given. Construction of close-to-convex harmonic functions is also studied by looking at transforms of convex analytic functions. Finally, a convolution property for harmonic functions is discussed. Key words: Harmonic, Convex, Close-to-Convex, Univalent. AMS subject classifications: 30C45, 58E Introduction Harmonic functions are famous for their use in the study of minimal surfaces also play important roles in a variety of problems in applied mathematics. Harmonic functions have been studied by differential geometers such as Choquet [2], Kneser [7], Lewy [8], Rado [9]. Recent interest in harmonic complex functions has been triggered by geometric functions theorists Clunie Sheil-Small [3]. A continuous function is a complex-valued harmonic functions in a domain if both are real harmonic in. In any simply connected domain, we can write where are analytic in. We call the analytic part the co-analytic part of. A necessary sufficient conditions (see [3] or [8] for to be locally univalent sensepreserving in is that in. Denote by the class of functions of the form (1 that are harmonic univalent sense-preserving in the unit disk for which. Thus we may write 1 Dedicated to KSU Professor Richard S. Varga on his seventieth birthday. Printed in the U.S.A by North Atlantic Science Publishing Company 23

2 24 JAY M. JAHANGIRI HERB SILVERMAN Note that reduces to, the class of normalized univalent analytic functions if the co- analytic part of is zero. Since for, the function is also in. Therefore, we may sometimes restrict ourselves to, the subclass of for which. In [3], it was shown that is normal is compact with respect to the topology of locally uniform convergence. Some coefficient bounds for convex starlike harmonic functions have recently been obtained by Avci Zlotkiewicz [1], Jahangiri [5, 6], Silverman [14]. In this paper, we give sufficient conditions for functions in to be close-to-convex harmonic or convex harmonic. We also construct close-to-convex harmonic functions by looking at transforms of convex analytic functions. Finally, we discuss a convolution property for harmonic functions. In the sequel, unless otherwise stated, we will assume that is of the form (1 with of the form (2. 2. Convex Close-to-Convex Mappings Let, denote the respective subclasses of, where the images of are convex. Similarly,, denote the subclass of, where the images of are close-to-convex. Recall that a domain is convex if the linear segment joining any two points of lies entirely in. A domain is called close-to-convex if the complement of can be written as a union of non-crossing half-lines. For other equivalent criteria, see [4]. Clunie Sheil-Small[3] proved the following results. Theorem A: If are analytic in with is close-toconvex for each,, then is harmonic close-to-convex. Theorem B: If is locally univalent in is convex for some, ", then is univalent close-to-convex. A domain is called convex in the direction " if every line parallel to the line through 0 has a connected intersection with. Such a domain is close-to-convex. The convex domains are those convex in every direction. We will also make use of the following result, which may be found in [3]. Theorem C: A function is harmonic convex if only if the analytic func- tions, ", are convex in the direction is suitably normalized. The harmonic Koebe function is defined by,, which leads to & & & & % %, The function maps onto the complex plane minus the real slit from % to. The coefficients of are % %. These coefficient bounds are known to be extremal for the subclass of consisting of typically real functions (e.g., see [3] functions that are either starlike or convex in one direction (e.g., see [12]. It is not known if the coefficients of are extremal for all of. Necessary coefficient conditions were found in [3] for functions to be in. We now give some sufficient condition for functions to be in these classes. But first we need the following results. See, for example, [13].

3 Harmonic Close-to-Convex Mappings 25 Lemma 1: If ' ( is analytic in, then ' maps onto a starlike domain if onto a convex domains if ( " ( " 3. Main Results Theorem 1: If with " & then. The result is sharp. Proof: In view of Theorem A, we need only prove that,, is close-toconvex. It suffices to show that Since " " if only if (3 holds, maps onto a starlike domain consequently. To see that the upper bound in (3 cannot be extended to,, we note that the function is not univalent in. Theorem 2: If is locally univalent with ", then. Proof: Take in Theorem B apply Lemma 1. Corollary: If " ", then. Proof: The function is locally univalent if for. Since " ", we have *. We next give a sufficient coefficient condition for to be convex harmonic. Theorem 3: If " + then. The result is sharp. Proof: By Theorem C, it suffices to show that is convex in. Set Since, " " if only if (4 holds, we see from Lemma 1 that, consequently. The function,, shows that the upper bound in (4 cannot be improved.

4 26 JAY M. JAHANGIRI HERB SILVERMAN Remark: The coefficient bound given in Theorem 3 can also be found in [5] [14]. However, our approach in this paper is different from those given in [5] [14]. Remark: The well-known results for univalent functions that is convex if only if is starlike does not carry over to harmonic univalent functions. See [12]. Hence, we cannot conclude from Theorem 3 that (3 is a sufficient condition for to map onto a starlike domain. Nevertheless, we believe this to be the case. See [5, 6, 14]. We now introduce a class of harmonic close-to-convex functions that are constructed from convex analytic functions. Theorem 4: If - is a Schwartz function, then -. Proof: Write -. Now for each,, we observe that / / -*, Consequently, is close-to-convex the result follows from Theorem A. Remark: If we only require that - in Theorem 4 be analytic with -,, then we may conclude that. Corollary: If is a positive integer, then.. Proof: A result of Sheil-Small [10] shows that. whenever. Set - in Theorem 4, the result follows. We now give some examples from Theorem 4. Example 1: Suffridge [15] showed for the partial sums 0 of 1 that Setting - in Theorem 4, we see that Example 2: Since 1, we get from the Corollary that & + + for &3 3. Example 3: Set - in Theorem 4. Then. / log We can actually state a more general result for which Example 3 is a special case.

5 Harmonic Close-to-Convex Mappings 27 Theorem 5: If is analytic with,, then. Proof: Set.. Then, so that is locally univalent. Set in Theorem B, the result follows. Corollary: If is analytic with " +,, then. 4. Convolution Condition The convolution of two harmonic functions 4 5 is defined by In [3], it was shown for that 6 ". We given 6 an example to show that cannot be replaced by 7, ", the family of functions starlike of order. Set 7,, 6 Then, see [3]. Setting in 6 we obtain which is not even univalent for. References [1] Avci, Y. Zlotkiewicz, E., On harmonic univalent mappings, Ann. Univ. Marie Curie-Sklodowska Sect. A 44 (1991, 1-7. [2] Choquet, G., Sur un type de transformation analytique generalisant la representation conforme et definie au moyen de fonctions harmoniques, Bull. Sci. Math. 69:2 (1945, [3] Clunie, J. Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A.I Math 9 (1984, [4] Duren, P.L., Univalent Functions, Springer-Verlag, New York 1983.

6 28 JAY M. JAHANGIRI HERB SILVERMAN [5] Jahangiri, J.M., Coefficient bounds univalence criteria for harmonic functions with negative coefficients, Ann. Univ. Marie Curie-Sklodowska Sect. A 52 (1998, [6] Jahangiri, J.M., Harmonic functions starlike in the unit disk, J. Math. Anal. Appl. 235 (1999, [7] Kneser, H., Lösung der aufgabe 41, Jahresber, Deutsch. Math.-Verein. 35 (1926, [8] Lewy, H., On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936, [9] Rado, T., Aufgabe 41, Jahresber. Deutsch. Math.-Verein. 35 (1926, 49. [10] Sheil-Small, T., On convex univalent functions, J. London Math. Soc. 2:1 (1969, [11] Sheil-Small, T., On linear accessibility the conformal mapping of convex domains, J. Analyse Math. 25 (1972, [12] Sheil-Small, T., Constant for planar harmonic mappings, J. London Math. Soc. 2:42 (1990, [13] Silverman, H., Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975, [14] Silverman, H., Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl. 220 (1998, [15] Suffridge, T.J., On a family of convex polynomials, Rocky Mountain J. Math. 22:1 (1992,

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