Growth and distortion theorem for the Janowski alpha-spirallike functions in the unit disc

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1 Stud. Univ. Babeş-Bolyai Math. 57(01), No., Growth and distortion theorem for the Janowski alpha-spirallike functions in the unit disc Yaşar Polato glu Abstract. Let A be the class of all analytic functions in the open unit disc D = < 1} of the form = + a + a Let g() be an element of A satisfying the condition e iα g () g() = 1 + Aφ() 1 + Bφ() where α < π, 1 B < A 1 and φ() is analytic in D and satisfies the conditions φ(0) = 0, φ() < 1 for every D. Then g() is called Janowski α-spirallike functions in the unit disc. The class of such functions is denoted by Sα(A, B). The aim of this paper is to give growth and distortion theorems for the class Sα(A, B). Mathematics Subject Classification (010): 30C45. Keywords: Growth theorem, distortion theorem, radius of starlikeness. 1. Introduction Let Ω be the family of functions φ() which are regular in the open unit disc D and satisfying the conditions φ(0) = 0, φ() < 1 for all D. Next, for arbitrary fixed numbers A, B, 1 B < A 1, denote by P(A, B), the family of functions p() = 1 + p 1 + p + regular in D, such that p() in P(A, B) if and only if p() = 1 + Aφ() (1.1) 1 Bφ() for some function φ() Ω, and for all D. At the same time, this class can be represented by Rep() > 1 A 1 B > 0. Let F () = + α + α and G() = + β + β be analytic functions in D. If there exists a function φ() Ω such that F () = G(φ()) for every D, then we say that F () is subordinate to G(), and we write F () G(). We also note that if F () G(), then F (D) G(D) ([1]).

2 56 Yaşar Polato glu Moreover, let Sα(A, B) denote the family of functions = + a + regular in D, such that is in Sα(A, B) if and only if there is a real number α for which, e iα f () = cos αp() + i sin α, α < π, p() P(A, B) (1.) is true for every D. Then the class Sα(A, B) is called the Janowski α-spirallike functions. The following lemma is due to I. S. Jack and plays very important role for our proof of Theorem.1 ([]). Lemma 1.1. Let φ() be regular in the unit disc D with φ(0) = 0. Then if φ() obtains its maximum value on the circle = r at the point 1, one has 1 φ ( 1 ) = kφ( 1 ), for some k 1.. Main results Theorem.1. S α(a, B) ( f () 1) e iα (A B) cos α. 1+B e iα (A cos α); B = 0, (.1) Proof. Let be an element of S α(a, B). We define the functions φ() by; = (A B) cos αe iα (1 + Bφ()) B e A cos αe iαφ() ; B = 0, (.) (A B) cos αe iα where (1 + Bφ()) B and e A cos αe iαφ() have the value 1 at = 0. Then φ() is analytic and φ(0) = 0. If we take the logarithmic derivative from (.) and after simple calculations, we get ( f () 1) = (A B) cos αe iα φ () 1+Bφ() A cos α.e iα φ (); B = 0, (.3) We can easily conclude that this subordination is equivalent to φ() < 1 for all D. On the contrary let s assume that there exists 1 D, such that φ() attains its maximum value on the circle = r, that is φ( 1 ) = 1. Then when the conditions 1 φ ( 1 ) = kφ( 1 ), k 1 are satisfied for such 1 D (Using I.S.Jack s Lemma), we obtain; ( 1 f ( 1 ) f( 1 ) 1) = (A B) cos αe iα kφ( 1) 1+Bφ( 1) = F 1 (φ( 1 )) / F 1 (D) A cos αe iα kφ( 1 ) = F (φ( 1 )) / F (D); B = 0, which contradicts (.1) implying that the assumption is wrong, i.e., φ() < 1 for all D. This shows that, Sα(A, B) ( f () (A B) cos αe iα 1) 1+B A cos α.e iα ; B = 0, (.4) (.5)

3 Growth and distortion theorem 57 Conversely, ( f () (A B) cos αe iα 1) 1+B A cos α.e iα ; B = 0, e iα f () = This shows that S α(a, B). cosα 1+Aφ() 1+Bφ() + i sin α cos α(1 + Aφ()) + i sin α; B = 0, Corollary.. Marx-Strohacker inequality for the class Sα(A, B) is; ( ) B.e iα (A B) cos α 1 < 1 log( ) e iα A cos α < 1; B = 0, (.6) Proof. The proof of this corollary is a simple consequence of Theorem.1. Indeed, = (1 + Bφ()) A B B cos αe iα ( ) B.e iα (A B) cos α 1 < 1 = e A cos αe iαφ() log( ) e iα A cos α < 1 Theorem.3. The radius of starlikeness of the class Sα(A, B) is, r = (A B) cos α+ ((A B) cos α+4[ab cos α+b sin α]) 1 A cos α ; B = 0, (.7) This radius is sharp because the extremal function is; (1 + B) A B B cos αe iα = e A cos αe iα ; B = 0, (.8) with ζ = r(r eiα ) 1 re iα and we obtain, ζ f (ζ) f(ζ) = 1 (A B) cos αr (AB cos α+b sin α)r 1 B r ; B 0, 1 Ar cos α; B = 0, (.9) Proof. Using (1.) we get; p() = 1 cos α (eiα f () i sin α) (.10) On the other hand, since p() P(A, B), then we have, 1 ABr p() (A B)r 1 B r 1 B r (.11)

4 58 Yaşar Polato glu The inequality (.11) was obtained by W. Janowski [5]. Using (.10) in (.11) and after straightforward calculations we get: 1 (A B) cos αr (AB cos α+b sin α)r 1 B r Re f () 1+(A B) cos αr (AB cos α+b sin α)r 1 B r (.1) 1 A cos αr Re f () 1 + A cos αr; B = 0, The inequalities (.1) shows that this theorem is true. Corollary.4. If we take A = 1, B = 1 we obtain, 1 r = cos α + sin α (.13) This is the radius of starlikeness of class of α-spirallike functions. This result was obtained independently and using different methods by both Robertson [4] and Libera [3]. We also note that if we give another special values to A and B, we obtain the radius of starlikeness of the subclass of α-spirallike functions. Corollary.5. Let be an element of S α(a, B), then Proof. Using (.1), (A B) cos α(cos α 1) r(1 Br) B (1 + Br) B (A B) cos α(cos α 1) r(1 Br) B (1 + Br) B re (A cos α)r re (A cos α)r ; B = 0 Re( f () ) = r log r and after the straightforward calculations we get the result. Also we note that these inequalities are sharp. Because the extremal function was given in Theorem.3. Corollary.6. If S α(a, B), then where [(1 Ar) cos α (1 Br) sin α]f (A, B, cos α, r) f () [(1 + Ar) cos α + (1 + Br) sin α]f (A, B, cos α, r) F (A, B, cos α, r) = (1 + Br) This inequality is sharp. B 1 (A B) cos α(cos α 1) (1 Br) B Proof. The proof of this corollary is based on the following observations i. p() P (A, B) 1 Ar 1+Ar 1 Br p() 1+Br ii. p() = 1 cos α (eiα f () i sin α), S α(a, B), p() P (A, B) iii. Corollary.5. using (i) and (ii) and after simple calculations we get: (1 Ar) cos α (1 Br) sin α 1 Br f () Considering (.14) and Corollary.5 we get the result. (1 + Ar) cos α + (1 + Br) sin α 1 + Br (.14)

5 References Growth and distortion theorem 59 [1] Goodman, A.W., Univalent functions, Volume I. Mariner Publishing, Tampa Florida, [] Jack, I.S., Functions starlike and convex of order α, J. London. Math. Soc., 3(1971), no.. [3] Libera, R.J., Univalent α-spiral functions, Canad. J. Math., 19(1967), [4] Robertson, M.S., Radii of star-likeness and close-to-convexity, Proc. Amer. Math. Soc., 14(1965), [5] Janowski, W., Some extremal problems for certain families of analytic functions, J. Ann. Polon. Math., 8(1973), Yaşar Polato glu Department of Mathematics and Computer Sciences İstanbul Kültür University, İstanbul, Turkey