Nunokawa, M. and Sokół, J. Osaka J. Math. 5 (4), 695 77 ON SOME LENGTH PROBLEMS FOR ANALYTIC FUNCTIONS MAMORU NUNOKAWA and JANUSZ SOKÓŁ (Received December 6, ) Let A be the class of functions Abstract z which are analytic in the unit disk {z Ï z }. Let C(r) be the closed curve which is the image of the circle z r under the mapping Û, L(r) the length of C(r), and let A(r) be the area enclosed by the curve C(r). It was shown in [3] that if f ¾ A, f is starlike with respect to the origin, and for r, A(r) A, an absolute constant, then (.) L(r) O ½ n a n z n as r. It is the purpose of this work to prove, using a modified methods than that in [3], a strengthened form of (.) for Bazilevi c functions, strongly starlike functions and for close-to-convex functions.. Introduction Let A be the class of functions (.) z which are analytic in the unit disk {z ¾ Ï z }. Let S denote the subclass of A consisting of all univalent in D. If f ¾ A satisfies Re z f ¼¼ (z) f ¼ (z) ½ n, a n z n z ¾ then is said to be convex in and denoted by ¾ K. Mathematics Subject Classification. Primary 3C45; Secondary 3C8.
696 M. NUNOKAWA AND J. SOKÓŁ If f ¾ A satisfies z f ¼ (z) Re, z ¾ then is said to be starlike with respect to the origin in and denoted by ¾S. Furthermore, If f ¾ A satisfies z f ¼ (z) (.) Re, z ¾ e i«g(z) for some g(z) ¾ S and some «¾ (, ), then is said to be close-to-convex in and denoted by ¾ C. An univalent function f ¾ S belongs to C if and only if the complement E of the image-region F { Ï z } is the union of rays that are disjoint (except that the origin of one ray may lie on another one of the rays). On the other hand, if f ¾ A satisfies Re z f ¼ (z) f (z)g (z), z ¾ for some g(z) ¾ S and some ¾ (, ½), then is said to be a Bazilevi c function of type and denoted by ¾ B( ). Let SS («) denote the class of strongly starlike functions of order «, «, SS («) Ï f ¾ AÏ Arg z f ¼ (z), «, z ¾ which was introduced in [] and []. Let C(r) be the closed curve which is the image of z r under the mapping Û. Let L(r) denote the length of C(r) and let A(r) be the area enclosed by C(r). Let us define M(r) by Then F.R. Keogh [4] has shown that M(r) max zr. Theorem.. Suppose that ¾ S and M ½, z ¾. Then we have L(r) O as r,
ON SOME LENGTH PROBLEMS 697 where O means Landau s symbol. Furthermore, D.K. Thomas in [3] extended this result for bounded close-to-convex functions. Ch. Pommerenke in [9] has shown that Theorem.. If ¾ C, then L(r) O M(r) 5 as r. Later, D.K. Thomas in [4] has shown that Theorem.3. If ¾ S, then ÔA(r) L(r) O as r. M. Nunokawa in [6, 7] has shown that Theorem.4. If ¾ K, then L(r) O A(r) as r. Moreover, D.K. Thomas in [5] has shown the following two theorems Theorem.5. Theorem.6. If ¾ B( ) and in, then we L(r) O as r. If ¾ B( ) and, then we L(r) O M(r) as r. M. Nunokawa, S. Owa et al. in [8] have shown that Theorem.7. If ¾ B( ) and z f ¼ (z) f (z)g (z)h(z), then we L(r) OÔA (r)g (r) as r,
698 M. NUNOKAWA AND J. SOKÓŁ where G(r) r ±g ¼ (±e i ) d d± or G(r) is the area of the image domain of z r under the starlike mapping g. Ch. Pommerenke in [9] has also shown that Theorem.8. If ¾ S, then Ö A(r) (.3) M(r) 4 3 (z r ). Therefore, we have M(r) O A(r) as r. It is the purpose of this work to prove, using a modified methods than that in [3], a strengthened form of (.) for Bazilevi c functions, strongly starlike functions and for close-to-convex functions.. Lemmas Lemma.. If h(z) is analytic and Re{h(z)} in with h(), then for r. h(re i ) d 3r 4 Lemma. can be easily proved using h (n) () n! and the Gutzmer s theorem, see for example [3, p. 3]. Lemma.. If ¾ S, then we have z f ¼ (z) z z z f ¼ (z) z ( z) 3 in. in, A proof can be found in [, p. ].
Lemma.3 ([, p. 337])., then we have (.) h ¼ (z) Re{h(z)} z z ON SOME LENGTH PROBLEMS 699 If h(z) is analytic and Re{h(z)} in with h() in. A proof can be found also in [5]. An analytic function f is said to be subordinate to an analytic function F, or F is said to be superordinate to f, if there exists a function an analytic function Û such that and Û() and Û(z) (z ¾ ), F(Û(z)) (z ¾ ). In this case, we write f F (z ¾ ) or F(z) (z ¾ ). If the function F is univalent in, then we have [ f F (z ¾ )] [ f () F() and f () F()]. Lemma.4. If is subordinate to g(z) in and if p, then for all r, r. f (re i ) p d g(re i ) p d W. Rogosinski has shown Lemma.4 in []. 3. Main results Theorem 3.. If ¾ S satisfies the condition (3.) Re z f ¼¼ (z) z Re z then we have (3.) L(r) O A(r) in, as r. Proof. For the case r, from Lemma. we have L(r). z f ¼ (z) d z( z) ( z) 3 d
7 M. NUNOKAWA AND J. SOKÓŁ For the case r, we have L(r) r r r z f ¼ (z) d f ¼ (z) z f ¼¼ (z) d± d f ¼ (z) r Ô A(r) z f ¼¼ (z) d± d r f ¼ (z) d± d ± f ¼ (z) d± d r From the hypothesis (3.), we have or (3.3) It follows that Re z f ¼¼ (z) r z f ¼¼ (z) d± d. z f ¼¼ (z) z f ¼¼ (z) ( z)( z) z f ¼¼ (z) z in z z z z z d± d z f ¼¼ (z) d± d z z in, where the symbol means the subordination. Then we have r r r z f ¼¼ (z) r r z f ¼¼ (z) z f ¼¼ (z) I I I 3. d d± z f ¼¼ (z) z z z d d± z z z d d± z z in. z z d d± z z d d±
ON SOME LENGTH PROBLEMS 7 From Lemma.4, (3.3) and Lemma., we have I By Lemma., we have I r O r r 3 r z f ¼¼ (z) 4 z z d d± ± d± 6 r. z f ¼¼ (z) 6 r 6 r z z 8 r 4 r as r. d d± z z d d± r ± d± z z d d± By Lemma., we have I 3 r 4 r O z z d d± as r. This shows (3.) which completes the proof of Theorem 3.. Theorem 3.. we have (3.4) L(r) O If ¾ B( ) is a Bazilevi c function of type,, then A(r) 3 as r. Proof. Because ¾ B( ), there exists g(z) ¾ S and there exists an analytic function h(z), h(), Re{h(z)} in, such that (3.5) z f ¼ (z) f (z)g (z)h(z).
7 M. NUNOKAWA AND J. SOKÓŁ Therefore we have L(r) M (r) M (r) z f ¼ (z) d f (z)g (z)h(z) d r g (z)h(z) d M (r)(i (r) I (r)). g (z)g ¼ (z)h(z) d d± r Applying Ch. Pommerenke s result (.3), we have 6 L(r) A(r) 3 ( ) (I (r) I (r)). g (z)h ¼ (z) d d± D.K. Thomas in [5] has shown that if is a Bazilevi c function of type,, then I (r) Ô K ( ) r r (3.6) O as r, where (3.7) K ( ) max{, (4r) } is a bounded constant not necessarily the same each time. On the other hand Using (.) we obtain I (r) I (r) r Re Using (3.5) we can write I (r) Re r r g (z)h ¼ (z) d d±. g(z) Re{h(z)} d d± ± g (z) g (z) g (z)h(z) ± d d±. r z f ¼ (z) f i arg g(z) e (z) ± d d±.
ON SOME LENGTH PROBLEMS 73 Because g(z) is a starlike function, then arg g(±e i ) is an increasing function of and maps the interval [, ] onto oneself. Applying D. K. Thomas method [5, p. 357], after a suitable substitution and integrating by parts, we obtain I (r) r d f Re (z) e i arg g(z) dz z dz ± iz d± Re Re Re r z± z± i r d± r r i ( ± ) d± i ( ± ) e d f (z) ± d arg g(z) i arg g(z) z± d arg g(z) d± e i arg g(z) d f (z) d arg g(z) [ f (z)e i arg g(z) ] Re z± r 4 M (±)( ± ) d±. r z± arg g(z) arg g(z) d arg g(z) i f (z)e i arg g(z) d arg g(z) f (z)e i arg g(z) ± d arg g(z) d± Applying Ch. Pommerenke s result (.3), we have I (r) 6 Ô r 3 A(±) ( ± ) d± ± 6 Ô r A 3 (r) ± ± d± 6 Ô A r 3 (r) ± O A ( ) (r) as r. Applying it together with (3.6) we obtain (3.4). Theorem 3.3. (3.8) L(r) O ( ) ¼ d± If ¾B( ) is a Bazilevi c function of type,, then we have A (r) as r.
74 M. NUNOKAWA AND J. SOKÓŁ Proof. For the case r, because B( ) S, by Lemma. we have L(r), z f ¼ (z) d r( r) () 3 d where r z. Assume that h(z) z f ¼ (z) f (z)g (z), Re{h(z)}, z ¾, g ¾ S. For the case r, we have L(r) z f ¼ (z) d f (z)g (z)h(z) d ( r) r g (z)h(z) d 9 g (z)h(z) d 9 r g ¼ (z)g (z)h(z) d± d 9 (I (r) I (r)). Using the result (3.7) for r, we have I (r) Ô K ( ) r, g (z)h ¼ (z) d± d where K ( ) max {, 8 }. Furthermore, in the same way as in the previous proof, we obtain I (r) r O g (z)h ¼ (z) d± d (A(r)) ( ) as r, where K (r) is a bounded function of. This completes the proof.
ON SOME LENGTH PROBLEMS 75 REMARK 3.4. D.K. Thomas in [5] has shown that if is a Bazilevi c function of type,, then L(r) K ( )M(r), where K ( ) is a bounded function of. On the other hand, from Ch. Pommerenke s result [9], we have Ô 3 L(r) K ( ) A(r). From Theorems 3. and 3.3 we have that if is a Bazilevi c function of type,, then O A (r) for, L(r) O Theorem 3.5.. Then we have A (r) (3.9) L(r) O 3 for, as r. Let f ¾ SS («) be strongly starlike function of order «, «A(r) as r. Proof. From the hypothesis of the Theorem and applying Ch. Pommerenke s [9] and Rogosinski s [] results, we have L(r) M(r) z f ¼ (z) d z f ¼ (z) d z f ¼ (z) d Ô K A(r) () Ô K A(r) () O A(r) z «z d z «d as r,
76 M. NUNOKAWA AND J. SOKÓŁ where K is a bounded constant and because we have d ½ for «. z «Corollary 3.6. Let f ¾ C be close-to-convex function, satisfy (.) with «in and map onto a domain of finite area A. Then by Theorem 3.,, we have L(r) O 3 as r. Notice that D.K. Thomas in Theorem [3, p. 43]. has shown that L(r) O as r. when f ¾ C, satisfies (.) with «and f is bounded in. Corollary 3.7. Let f ¾ C be close-to-convex function, satisfy (.) with «in. Then by Theorem 3.,, we have L(r) O A(r) 3 as r. In [3] it was shown that L(r) O M(r) as r, when f ¾ C, satisfies (.) with «. Compare also Theorems..8 in the introduction. References [] D.A. Brannan and W.E. Kirwan: On some classes of bounded univalent functions, J. London Math. Soc. () (969), 43 443. [] G. Goluzin: Geometrische Functionentheorie, V.E.B. Deutscher Verlag der Wissenscheften, Berlin 957. [3] A.W. Goodman: Univalent Functions, I, II Mariner, Tampa, FL, 983. [4] F.R. Keogh: Some theorems on conformal mapping of bounded star-shaped domains, Proc. London Math. Soc. (3) 9 (959), 48 49. [5] T.H. MacGregor: The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 4 (963), 54 5.
ON SOME LENGTH PROBLEMS 77 [6] M. Nunokawa: On Bazilevič and convex functions, Trans. Amer. Math. Soc. 43 (969), 337 34. [7] M. Nunokawa: A note on convex and Bazilevič functions, Proc. Amer. Math. Soc. 4 (97), 33 335. [8] M. Nunokawa, S. Owa, T. Hayami and K. Kuroki: Some properties of univalent functions, Int. J. Pure Appl. Math. 5 (9), 63 69. [9] Ch. Pommerenke: Über nahezu konvexe analytische Funktionen, Arch. Math. (Basel) 6 (965), 344 347. [] Ch. Pommerenke: Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 975. [] W. Rogosinski: On the coefficients of subordinate functions, Proc. London Math. Soc. () 48 (943), 48 8. [] J. Stankiewicz: Quelques problèmes extrémaux dans les classes des fonctions «-angulairement étoilées, Ann. Univ. Mariae Curie Skłodowska Sect. A (966), 59 75. [3] D.K. Thomas: On starlike and close-to-convex univalent functions, J. London Math. Soc. 4 (967), 47 435. [4] D.K. Thomas: A note on starlike functions, J. London Math. Soc. 43 (968), 73 76. [5] D.K. Thomas: On Bazilevič functions, Trans. Amer. Math. Soc. 3 (968), 353 36. Mamoru Nunokawa University of Gunma Hoshikuki-cho 798-8 Chuou-Ward, Chiba, 6-88 Japan e-mail: mamoru_nuno@doctor.nifty.jp Janusz Sokół Department of Mathematics Rzeszów University of Technoy Al. Powstańców Warszawy 35-959 Rzeszów Poland e-mail: jsokol@prz.edu.pl