Banach Journal of Mathematical Analysis ISSN: (electronic)

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1 Banach J. Math. Anal. (008), no., 16 Banach Journal of Mathematical Analysis ISSN: (electronic) GENEALIZATION OF SǍLǍGEAN OPEATO FO CETAIN ANALYTIC FUNCTIONS ALAWIAH IBAHIM 1, SHIGEYOSHI OWA, MASLINA DAUS 3 AND YAYOI NAKAMUA 4 This paper is dedicated to Professor Josip E. Pečarić Submitted by M. Joita Abstract. For analytic functions f in the open unit disc U, a generaliation operator D λ f() of Sǎlǎgean operator is introduced. Some properties for D λ f() are discussed in the present paper. 1. Introduction and preliminaries Let A be the class of functions f of the form f() + a k k which are analytic in the open unit disk U { C : 1}. For f A, Sǎlǎgean[5] has defined the following operator D n f() by (i) (ii) (iii) D 0 f() f() D 1 f() Df() f () + ka k k, D n f() D(D n 1 f()) + kn a k k, (n 1,, 3...). Date: eceived: 8 February 008; Accepted: 9 April 008. Corresponding author. 000 Mathematics Subject Classification. 30C45. Key words and phrases. Sǎlǎgean operator, analytic function, starlike function. 16

2 SǍLǍGEAN OPEATO FO CETAIN ANALYTIC FUNCTIONS 17 In view of the Sǎlǎgean operator, we introduce D λ f() + k λ a k k, (λ ) for f A. Then for any real λ we see that + k λ+1 a k k (D λ f() and It is easy to see that for any real λ 1 and λ. D λ 1 f() + k λ 1 a k k 0 D λ f(t) dt. t D λ 1+λ f() D λ (D λ 1 f()) D λ 1 (D λ f()) To discuss our new problem, we have to recall here the following lemma by Jack [1] (also by Miller and Mocanu [3]). Lemma 1.1. Let w() be non-constant and analytic in U with w(0) 0. If w() attains its maximum value on the circle r at the point 0 U, then we have 0 w( 0 kw( 0 ) where k 1 is real.. Properties of the operator D λ f() Our first result for the operator D λ f() is contained in the following theorem. Theorem.1. If f A satisfies ( D λ f() for some real α, with α + 0 and for any real λ, then ( ) > 0 ( U). D λ f() Proof. Let us define w() by D λ f() 1 + w() 1 w() Then w() is analytic in U and w() 0. Since ( ) 1 ( U) (.1) (w() 1). ( ) D λ+1 f() w () D λ f() (1 w()),

3 18 A. IBAHIM, S. OWA, M. DAUS & Y. NAKAMUA we obtain that D λ f() ( D λ f() α w() w( 1 w() (1 w()) ( ) 1 for all U. If there exists a point 0 U such that max 0 w() w( 0 ) 1, then Lemma 1.1 gives us that w( 0 ) e iθ and 0 w ( 0 ) ke iθ (k 1). This implies that D λ+1 f( 0 ) α ( ) 1 D λ+1 f( 0 ) D λ f( 0 ) 0 D λ f( 0 ) e iθ α ke iθ 1 e iθ (1 e iθ ) ( ) α+ k k 1 e iθ α+ ( ) 1 for all U, which contradicts the condition of the theorem. This show that there is no 0 U such that w( 0 ) 1. Therefore w() 1 for all U which implies that ( ) > 0 ( U). D λ f() This completes the proof of the theorem. Noting that if f A is starlike in U which is equivalent to ( ) f () > 0 ( U), f() then a k k (k, 3, 4,...) and equality holds true for Koebe function f given by f() the extremal function for the class of starlike functions in U. Thus we have which is (1 ) Corollary.. If f A satisfies the inequality (.1) for some real α, with α + 0 and for any real λ, then Equality holds true for Koebe function. a k k 1 λ (k, 3, 4,...). By the Marx-Strohhäcker theorem ([], [6]), we know that if f A satisfies ( 1 + f ) () > 0 ( U), f () then ( ) f () > 1 ( U). f() If we define the function F () by F () D λ 1 f(), then F () D λ f() and F () + F (). Therefore, we have

4 SǍLǍGEAN OPEATO FO CETAIN ANALYTIC FUNCTIONS 19 Corollary.3. If f A satisfies the inequality (.1) for some real α, with α + 0 and for any real λ, then ( ) D λ f() > 1 ( U). D λ 1 f() The result is sharp for the function f given by f() + k 1 λ k which is equivalent to D λ f() Next we prove the following theorem. (1 ). Theorem.4. If f A satisfies ( D λ f() ( ) 1 (1 γ) α+ ( U) (.) for some real α,, γ with α + 0 and 0 γ 1, then ( ) > γ ( U). D λ f() Proof. Defining the function w() by D λ f() 1 + (1 γ)w() 1 w() (w() 1), we see that w() is analytic in U and w(0) 0. Note that ( ) D λ+1 f() (1 γ)w (). D λ f() (1 w()) Thus we have that D λ f() ( D λ f() α (1 γ)w() (1 γ)w( 1 w() (1 w()) ( ) 1 (1 γ) α+ ( U). If there exists a point 0 U such that max 0 w() w( 0 ) 1, then w() satisfies w( 0 ) e iθ and 0 w ( 0 ) ke iθ (k 1) by Lemma 1.1.

5 0 A. IBAHIM, S. OWA, M. DAUS & Y. NAKAMUA This gives us that D λ+1 f( 0 ) α ( ) 1 D λ+1 f( 0 ) D λ f( 0 ) 0 D λ f( 0 ) (1 γ)e iθ α (1 γ)ke iθ 1 e iθ (1 e iθ ) α+ k (1 γ) α+ 1 e iθ α+ ( ) k (1 γ) α+ ( ) 1 (1 γ) α+ ( U) which contradicts the condition of the theorem. This show that there is no 0 U such that w( 0 ) 1. Therefore w() 1 for all U. Thus we conclude that ( ) > γ ( U). D λ f() Noting that if f A satisfies ( ) f () > γ ( U), f() then a k k j (j γ) (k 1)! (k, 3, 4,...) and equality holds true for the functions f given by f() (1 ) (1 γ) which is the extremal function for the class of starlike of order γ in U (cf. obertson[4]). In view of the above, we give direct corollary as follows: Corollary.5. If f A satisfies the inequality (.) for some real α,, γ with α + 0 and 0 γ 1, then k j (j γ) a k (k, 3, 4,...). k λ (k 1)! Equality holds true for the function f given by k j (j γ) f() + k λ (k 1)! which is equivalent to D λ f() (1 ) (1 γ). k

6 SǍLǍGEAN OPEATO FO CETAIN ANALYTIC FUNCTIONS 1 Finally, we derive the following: Theorem.6. If f A satisfies ( D λ f() ( γ ) ( U) for some real α,, and γ α +, then ( ) 1 γ > 0 ( U). D λ f() Proof. Defining the function w() by D λ f() ( ) γ 1 + w() (w() 1) 1 w() with γ, we see that w() is analytic in U and w(0) 0. Noting that α + ( ) D λ+1 f() γw () D λ f() (1 w()) ( ) γ w(), 1 w() we have that ( D λ f() α+(γ 1) 1 + w() γw( 1 w() (1 w()) γw( (1 w()) ( γ ) ( U) since γ α +. Now, suppose that there exists a point 0 U such that max 0 w() w( 0 ) 1. Then, by Lemma 1.1, we have that w( 0 ) e iθ and 0 w ( 0 ) ke iθ (k 1).

7 A. IBAHIM, S. OWA, M. DAUS & Y. NAKAMUA This gives us that D λ+1 f( 0 ) α ( ) 1 D λ+1 f( 0 ) D λ f( 0 ) 0 D λ f( 0 ) γke iθ (1 e iθ ) k γ (1 e iθ ) ( ) kγ ( γ ) ( U) which contradicts the condition of the theorem. This show that there is no 0 U such that w( 0 ) 1. Therefore, we conclude that w() 1 for all U, that is, that ( ) 1 γ > 0 ( U). D λ f() Acknowledgements: The work presented here was fully supported by escience- Fund: SF045, MOSTI, Malaysia and completed at Universiti Kebangsaan Malaysia when the second author has visited in Malaysia. eferences 1. I.S. Jack, Functions starlike and convex of order α, J. London Math. Soc., 3 (1971), A. Marx, Untersuchungen uber schlichte Abbildungen, Math. Ann., 107 (193/33), S.S. Miller and P.T. Mocanu, Second-order differential inequalities in complex plane, J. Math. Anal. Appl., 65 (1978), M.S. obertson,on the theory of univalent functions, Ann. Math., 37 (1936), G.S. Sǎlǎgean, Subclasses of univalent functions, Lecture Notes in Math., 1013, Springer- Verlag, Berlin, 1983, E. Strohhäcker, Beitrage ur Theorie der schlichten Funktionen, Math. Z., 37 (1933), ,3 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor D. Ehsan, Malaysia. address: 1 alaibra@ukm.my, 3 maslina@ukm.my,4 Department of Mathematics, Kinki University, Higashi-Osaka, Osaka , Japan. address: owa@math.kindai.ac.jp, 4 yayoi@math.kindai.ac.jp