Applied Mathematics Letters 3 (00) 777 78 Contents lists availale at ScienceDirect Applied Mathematics Letters journal homepage: wwwelseviercom/locate/aml Fekete Szegö prolem for starlike convex functions of complex order S Kanas a,, HE Darwish a Department of Mathematics, Rzeszow University of Technology, W Pola, PL-35-959 Rzeszów, Pol Department of Mathematics, Faculty of Sciences, Mansoura University, Mansoura 3556, Egypt a r t i c l e i n f o a s t r a c t Article history: Received 3 July 009 Received in revised form 7 Septemer 009 Accepted March 00 For nonzero complex let F n () denote the class of normalized univalent functions f satisfying Re + (z(d n f ) (z)/d n f (z) )/] > 0 in the unit disk U, where D n f denotes the Ruscheweyh derivative of f Sharp ounds for the Fekete Szegö functional a3 µa are otained 00 Elsevier Ltd All rights reserved Keywords: Coefficient estimates Ruscheweyh derivative Fekete Szegö prolem Convex starlike functions of complex order Introduction Fekete Szegö proved a noticeale result that the estimate ( ) λ a 3 λa + exp λ holds for any normalized univalent function f (z) z + a z + a 3 z 3 + () in the open unit disk U for 0 λ This inequality is sharp for each λ (see ]) The coefficient functional (f (0) 3λ ) f (0)] Φ(f ) a 3 λa 6 on normalized analytic functions f in the unit disk represents various geometric quantities, for example when λ, Φ(f ) a 3 a, ecomes S f (0)/6 where S f denotes the Schwarzian derivative (f /f ) (f /f ) / Note that, if we consider the nth root transform f (z n )] /n z + c n+ z n+ + c n+ z n+ + of f with the power series (), then c n+ a /n c n+ a 3 /n + (n )a /n, so that a 3 λa n ( c n+ µc n+), Corresponding author E-mail addresses: skanas@przrzeszowpl (S Kanas), darwish333@yahoocom (HE Darwish) 0893-9659/$ see front matter 00 Elsevier Ltd All rights reserved doi:006/jaml0003008
778 S Kanas, HE Darwish / Applied Mathematics Letters 3 (00) 777 78 where µ λn + (n )/ Moreover, Φ(f ) ehaves well with respect to the rotation, namely Φ(e iθ f (e iθ z)) e iθ Φ(f ), θ R This is quite natural to discuss the ehavior of Φ(f ) for suclasses of normalized univalent functions in the unit disk This is called Fekete-Szegö prolem Actually, many authors have considered this prolem for typical classes of univalent functions (see, for instance 7,,8 ]) We denote y S the set of all functions normalized analytic univalent in the unit disk U of the form () Also, for 0 α <, let S (α) S c (α) denote classes of starlike convex univalent functions of order α, respectively, ie S (α) f (z) S : Re zf (z) f (z) > α, z U ( S c (α) f (z) S : Re + zf ) (z) > α, f z U (3) (z) A notions of α-starlikeness α-convexity were generalized onto a complex order α y Nasr Aouf 3], Wiatrowski 4], Nasr Aouf 5] Oserve that S (0) S S c (0) S c represent stard starlike convex univalent functions, respectively Let f (z) z + k a kz k g(z) z + k kz k e analytic functions in U The Hadamard product (convolution) of f g, denoted y f g is defined y (f g)(z) z + a k k z k, z U k Let n N 0 0,,, The Ruscheweyh derivative of the n th order of f, denoted y D n f (z), is defined y D n f (z) z(zn f (z)) (n), n N 0 n! Ruscheweyh 6] determined that D n f (z) z ( z) f (z) z + Γ (n + k) n+ Γ (n + )(k )! a kz k (4) k The Ruscheweyh derivative gave an impulse for various generalization of well known classes of functions Exemplary, for α (0 α < ) n N 0, Ahuja 7,8] defined the class of functions, denoted R n (α), which consists of univalent functions of the form () that satisfy the condition Re z(dn f (z)) D n f (z) > α, z U We note that R 0 (α) S (α) R (α) S c (α) The class R n (0) R n was studied y Singh Singh 9] With the aid of Ruscheweyh derivative Kumar et al 0] introduced the class F n () of function f S as follows: Definition (0]) Let e a nonzero complex numer, let f e an univalent function of the form (), such that D n f (z) 0 for z U \ 0 We say that f elongs to F n () if Re + ( z(d n f (z)) ) > 0, z U (6) D n f (z) By giving specific values to n, we otain the following important suclasses studied y various researchers in earlier works, for instance, F 0 () S ( ) (Nasr Aouf 3]), F () S c ( ) (Wiatrowski 4], Nasr Aouf 5]) Moreover, when α 0, ) F n ( α) R n (α) (Singh Singh 9], Darus Akarally ]) Main results We denote y P a class of the analytic functions in U with p(0 ) Re p(z) > 0 We shall require the following: Lemma (], p 66) Let p P with p(z) + c z + c z +, then c n, for n If c then p(z) p (z) ( + γ z)/( γ z) with γ c / Conversely, if p(z) p (z) for some γ, then c γ c Furthermore we have c c c () (5)
If c < c c c, then p(z) p (z), where p (z) + z γ z+γ + γ γ z z γ z+γ, + γ γ z S Kanas, HE Darwish / Applied Mathematics Letters 3 (00) 777 78 779 γ c /, γ c c 4 c Conversely if p(z) p (z) for some γ < γ, then γ c /, γ c c c c c Theorem Let n 0 let e nonzero complex numer If f of the form () is in F n (), then a (n + ), 4 c a 3 max, +, () (n + )(n + ) a 3 n + n + a (n + )(n + ) Equality in () holds if z(d n f (z)) /(D n f (z)) + p (z) ], in () if z(d n f (z)) /(D n f (z)) + p (z) ], where p, p are given in Lemma Proof Denote F(z) D n f (z) z + A z + A 3 z 3 +, then (n + )(n + ) A (n + )a, A 3 a 3 (3) By the definition of the class F n () there exists p P such, that zf (z) F(z) z( + A z + 3A 3 z + ) z + A z + A 3 z 3 + which implies the equality + ( + c z + c z + ), + p(z), so that z + A z + 3A 3 z 3 + z + (A + c )z + (A 3 + c A + c )z 3 + (c A 3 + c A + c 3 + A 4 )z 4 + Equating the coefficients of oth sides we have A c, A 3 ( ) c c ( + ) + c 4, (4) so that, on account of (4) a n + c, a 3 (n + )(n + ) Taking into account (5) Lemma, we otain a n + c n +, a 3 c c (n + )(n + ) + + c] c ] + + c (n + )(n + ) ] + + c (n + )(n + ) max, + + ] (n + )(n + ) () ] c + c (5) (6)
780 S Kanas, HE Darwish / Applied Mathematics Letters 3 (00) 777 78 Thus a 3 max, + (n + )(n + ) Moreover a 3 n + n + a ( c + c (n + )(n + ) ) c n + (n + ) n + c (n + )(n + ) (n + )(n + ), as asserted Remark In the aove Theorem a special case of Fekete-Szegö prolem eg for real µ (n + )/(n + ) occurred very naturally simple estimate was otained Now, we consider functional a3 µa for complex µ Theorem 3 Let e a nonzero complex numer let f F n () Then for µ C a3 µa (n + )(n + ) max n, + + µ n + For each µ there is a function in F n () such that equality holds Proof Applying (5) we have a 3 µa c + c (n + )(n + ) ] c µ (n + ) ] c + c µ(n + ) (n + )(n + ) (n + ) c c c (n + )(n + ) + c ( + µ n + n + )] Then, with the aid of Lemma, we otain a3 µa c (n + )(n + ) + c ] n + + µ n + ( + c + µ n + )] (n + )(n + ) n + (n + )(n + ) max n, + + µ n + An examination of the proof shows that equality is attained for the first case, when c 0, c, then the functions in F n () is given y z(d n f (z)) D n f (z) + ( )z, z, for the second case, when c c, so that z(d n f (z)) D n f (z) respectively + ( )z z, (7) (8) We next consider the case, when µ are real Then we have: Theorem 4 Let > 0 let f F n () Then for µ R we have
( + (n + )(n + ) a3 µa (n + )(n + ) (n + )(n + ) S Kanas, HE Darwish / Applied Mathematics Letters 3 (00) 777 78 78 µ n + n + µ n + n + ] )] if µ n + n +, if n + (n + )( + ) µ, n + (n + ) (n + )( + ) if µ (n + ) For each µ there is a function in F n () such that equality holds Proof First, let µ n+ (+)(n+) In this case (5) Lemma give n+ (n+) a3 µa c (n + )(n + ) + c ( + µ n + )] n + ( + µ n + )] (n + )(n + ) n + Let, now n+ µ (+)(n+) Then, using the aove calculations, we otain n+ (n+) a3 µa (n + )(n + ) Finally, if µ (+)(n+), then (n+) a3 µa c (n + )(n + ) + c ( µ n + )] n + + c ( µ n + )] (n + )(n + ) n + µ n + ] (n + )(n + ) n + Equality is attained for the second case on choosing c 0, c in (7) in (8) c, c, c i, c for the first third case, respectively Thus the proof is complete Remark (i) Setting α in the aove results, we get the results from ] As an analogue to the complex nth starlikeness of a complex order we may introduce the notion of nth convexity of a complex order as follows: Definition Let e a nonzero complex numer, let f e an univalent of the form () We say that f elongs to S c n () if Re + z(d n f (z)) > 0, (D n f (z)) z U (9) Using the well known Alexer relation f S c zf S we easily otain ounds of coefficients a solution of the Fekete Szegö prolem in S c n () Theorem 5 Let e a nonzero complex numer let f S c n () If f of the form () is in Sc n (), then a (n + ), a 3 ( + ) 3(n + )(n + ), a 3 4 n + 3 n + a 3(n + )(n + )
78 S Kanas, HE Darwish / Applied Mathematics Letters 3 (00) 777 78 Reasoning in the same line as in the proof of Theorem 3 we otain Theorem 6 Let e a nonzero complex numer let f S c n () Then, for µ C holds a3 µa 3(n + )(n + ) max 3 n, + + µ n + For each µ there is a function in F n () such that equality holds References ] M Fekete, G Szegö, Eine Bemerkung üer ungerade schlichte Funktionen, J Lond Math Soc 8 (933) 85 89 ] HR Adel-Gawad, DK Thomas, The Fekete Szegö prolem for strongly close-to-convex functions, Proc Amer Math Soc 4 (99) 345 349 3] HS Al-Amiri, Certain generalization of prestarlike functions, J Aust Math Soc 8 (979) 35 334 4] JH Choi, YCh Kim, T Sugawa, A general approach to the Fekete Szegö prolem, J Math Soc Japan 59 (3) (007) 707 77 5] A Chonweerayoot, DK Thomas, W Upakarnitikaset, On the Fekete Szegö theorem for close-to-convex functions, Pul Inst Math (Beograd) (NS) 66 (99) 8 6 6] M Darus, DK Thomas, On the Fekete Szegö theorem for close-to-convex functions, Math Japonica 44 (996) 507 5 7] M Darus, DK Thomas, On the Fekete Szegö theorem for close-to-convex functions, Math Japonica 47 (998) 5 3 8] S Kanas, A Lecko, On the Fekete Szegö prolem the domain convexity for a certain class of univalent functions, Folia Sci Univ Tech Resov 73 (990) 49 58 9] FR Keogh, EP Merkes, A coefficient inequality for certain classes of analytic functions, Proc Amer Math Soc 0 (969) 8 0] W Koepf, On the Fekete Szegö prolem for close-to-convex functions, Proc Amer Math Soc 0 (987) 89 95 ] RR London, Fekete Szegö inequalities for close-to-convex functions, Proc Amer Math Soc 7 (993) 947 950 ] W Ma, D Minda, A unified treatment of some special classes of univalent functions, in: Z Li, F Ren, L Yang, S Zhang (Eds), Proceeding of Conference on Complex Analytic, Int Press, 994, pp 57 69 3] MA Nasr, MK Aouf, Starlike function of complex order, J Natur Sci Math 5 (985) 4] P Wiatrowski, The coefficients of a certain family of holomorphic functions, Zeszyty Nauk Uniw Lodz, Nauki Mat Przyrod Ser II (97) 75 85 5] MA Nasr, MK Aouf, On convex functions of complex order, Mansoura Sci Bull (98) 565 58 6] S Ruscheweyh, New criteria for univalent functions, Proc Amer Math Soc 49 (975) 09 5 7] OP Ahuja, Integral operator of certain univalent functions, Int J Math Sci 8 (4) (985) 653 66 8] OP Ahuja, On the radius prolems of certain analytic functions, Bull Korean Math Soc () (985) 3 36 9] R Singh, S Singh, Integrals of certain univalent functions, Proc Amer Math Soc 77 (979) 336 340 0] V Kumar, SL Shukla, AM Chaudhary, On a class of certain analytic functions of complex order, Tamkang J Math () (990) 0 09 ] M Darus, A Akarally, Coefficient estimates for Ruscheweyh derivative, Int J Math Math Sci (004) 937 94 ] C Pommerenke, Univalent Functions, in: Studia Mathematica Mathematische Lehrucher, Venhoeck Ruprecht, 975