Coefficients of the Inverse of Strongly Starlike Functions

Transcription

1 BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Bull. Malaysian Math. S. So. (Seond Series) 6 (00) 6 7 Coeffiients of the Inverse of Strongly Starlie Funtions ROSIHAN M. ALI Shool of Mathematial Sienes Universiti Sains Malaysia 800 USM Pulau Pinang Malaysia rosihan@s.usm.my Dediated to the memory of Professor Mohamad Rashidi Md. Raali Abstrat. For the lass of strongly starlie funtions sharp bounds on the first four oeffiients of the inverse funtions are determined. A sharp estimate for the Feete-Segö oeffiient funtional is also obtained. These results were dedued from non-linear oeffiient estimates of funtions with positive real part.. Introdution An analyti funtion f in the open unit dis U = { : < } is said to be strongly starlie of order 0 < if f is normalied by f ( 0) = 0 = f (0) and satisfies π arg < f ( ( U ). The set of all suh funtions is denoted by SS * ( ). This lass has been studied by several authors [ ]. In [5] it was shown that an univalent funtion f belongs to SS * ( ) if and only if for every w f (U ) a ertain lens-shaped region with verties at the origin O and w is ontained in f (U ). If f ( = + a + a + () is in the lass SS * ( ) the inverse of f admits an expansion f ( w) = w + + w w + ()

2 6 R.M. Ali near w = 0. It is our purpose here to determine sharp bounds for the first four oeffiients of and to obtain a sharp estimate for the Feete-Segö oeffiient funtional t. n. Preliminary results Let us denote by P the lass of normalied analyti funtions p in the unit dis U with positive real part so that p ( 0) = and Re p( > 0 U. It is lear that if and only if there exists a funtion p P so that / f ( = p (. By equating oeffiients eah oeffiient of terms of oeffiients of a funtion For example f ( = + a + a + f SS ) an be expressed in p ( = in the lass P. * ( a a a = = = + + () Using representations () and () together with f ( f ( w)) = w or w = f ( w) + a ( f ( w)) + a ( f ( w)) + we obtain the relationships = a = a + a = a + aa 5 5 a () Thus oeffiient estimates for the lass SS * ( ) and its inverses beome non-linear oeffiient problems for the lass P. Our prinipal tool is given in the following lemma.

3 Coeffiients of the Inverse of Strongly Starlie Funtions 65 Lemma []. A funtion p ( ) = + = belongs to P if and only if j= 0 j + = + j = j 0 / for every sequene { } of omplex numbers whih satisfy lim sup <. Lemma. If = p ( = + P μ { μ } = max μ 0 μ elsewhere If μ < 0 or μ > p ( = ( + ε ( ε ε =. If 0 < μ < p ( = ( + ε ) ( ε ) ε =. For μ = 0 + ε ε p ( : = p ( = λ + ( λ) 0 λ ε =. ε + ε For μ = p is the reiproal of. Remar. Ma and Minda [6] had earlier proved the above result. We give a different proof. Proof. Choose the sequene { } = μ and = 0 if >. This yields 0 that is = p of omplex numbers in Lemma to be μ + ( μ) + μ + μ ( μ ). (5)

4 66 R.M. Ali If μ < 0 or μ > the expression on the right of inequality (5) is bounded above by ( μ ). Equality holds if and only if i.e. p( = ( + ( or its rotations. If 0 < μ < the right expression of inequality (5) is bounded above by. In this ase = 0 and = i.e. p( = ( + ) ( ) or its rotations. Equality holds when μ = 0 if and only if i.e. [8 p. ] = = + ε ε p ( : = p ( = λ + ( λ) 0 λ ε =. ε + ε = Finally when μ = if and only if p is the reiproal of. to be find that Another interesting appliation of Lemma ours by hoosing the sequene 0 δ β p { } = = and = 0 if >. In this ase we = β + ) + δ + ( ) + (δ ) (β ) ( ( δ β ) β ( β ) ν = + ( ) + β ( β ) (6) δ ( β ) + β ( δ ) where ν : =. β ( β ) Lemma. Let = p ( = + P. If 0 β and β ( β ) δ β β + δ. Proof. If β = 0 δ = 0 and the result follows sine. If β = δ = and the inequality follows from a result of []. We may assume that 0 < β < so that β ( β ) < 0. With = β we find from (6) that

5 Coeffiients of the Inverse of Strongly Starlie Funtions 67 ν β + δ + β ( β ) + β ( β ) ( δ β ) β ( β ) + bx + x : = h( x) with x = [0 ] b = β ( β ) and = ( δ β ) β ( β ). Sine 0 it follows that h( x) h(0) provided h ( 0) h() 0 i.e. b + 0. This ondition is equivalent to δ β β ( β ) whih ompletes the proof. With δ = β in Lemma we obtain an extension of Libera and Zlotiewi [] result that +. = Corollary. If p ( = + P and 0 β β + β. When β = 0 πi / + εe ( = λ πi / = εe p( : = p ε ( = ) λ 0 with λ + λ + λ. If β = p is the = reiproal of p. If 0 < β < p ( = ( + ε ε ε = or p ( = ( + ε ) ( ε ) ε =. Proof. We only need to find the extremal funtions. If β = 0 equality holds if and only if i.e. p is the funtion p [8 p. ]. If β = equality holds = if and only if p is the reiproal of p When 0 < β < we dedue from (6) that. β + β + β ( β ) + β ( β ) β ( β ) + β ( β ) β ( β ).

6 68 R.M. Ali Equality ours in the last inequality if and only if either = 0 or =. If 0 0 i.e. p ( = ( + ε ) ( ε ) ε =. If = = p ( = ( + ε ( ε ε =. = Lemma. If = p ( = + P { μ } = ( μ + ) + μ max μ 0 μ elsewhere Proof. For 0 μ the inequality follows from Lemma with δ = μ and β = μ +. For the seond estimate hoose β = μ = and δ = μ in (6). Sine μ ( μ ) > 0 we onlude from (5) and (6) that ( μ + ) + μ + μ( μ ) (μ ).. Coeffiient bounds Theorem. Let f SS * ( ) and f ( w) = w + w + w +. Then with equality if and only if f ( + ε = ε ε =. (7) Further 5 0 < 5 5 For > / 5 extremal funtions are given by (7). If 0 < < / 5 equality holds if and only if f ( + ε = ε ε = (8)

7 Coeffiients of the Inverse of Strongly Starlie Funtions 69 while if = / 5 f ( Moreover = + ε ε p ( = λ + ( λ) ε = 0 λ. ε + ε 9 ( 6 + ) 0 < For / extremal funtions are given by (7) while for 0 < f ( + ε = ε ε =. Proof. The following relations are obtained from () and (): = + 5 = (9) = ( + 5 ) + : = E 6 The bound on follows immediately from the well-nown inequality. Lemma with μ = + 5 yields the bound on and the desription of the extremal funtions. For the fourth oeffiient we shall apply Lemma with β = + 5 and δ = ( ) 6. The onditions on β and δ are satisfied if. with equality if and only if f ( / f ( = [( + ε ) ( ε )]. Thus For < / 5 Corollary yields E + 5 ( + 5 ) + + ( 6 + ). 6

8 70 R.M. Ali It remains to determine the estimate for / 5 <. Appealing to Lemma with μ = 5 and beause 5 + > 0 in (0 ] we onlude that E 5 + ( + 5 ) (0 ) 6 + ( 5 + ) = ( 6 + ). Theorem. Let f SS * ( ) and f ( w) = w + w + w +. Then t (5 t) (t 5) 5 t 5 t 5 + t / 5 + / < 5 / f If < t is given by (8). If t < or 5 + / 5 + / t > f is given by (7). If t = equality holds ( f ( 5 / if and only if = p while if t = = p (. f ( Proof. From (9) we obtain The result now follows from Lemma. + (5 t) t =. Remar. An equivalent result for the Feete-Segö oeffiient funtional over the lass SS * ( ) was also given by Ma and Minda [5]. Anowledgement. This researh was supported by a Universiti Sains Malaysia Fundamental Researh Grant. The author is greatly indebted to Professor R.W. Barnard for his helpful omments in the preparation of this paper.

9 Coeffiients of the Inverse of Strongly Starlie Funtions 7 Referenes. D.A. Brannan J. Clunie and W.E. Kirwan Coeffiient estimates for a lass of starlie funtions Canad. J. Math. (970) D.A. Brannan and W.E. Kirwan On some lasses of bounded univalent funtions J. London Math. So. (969).. C.R. Leveren Hermitian forms in funtion theory Trans. Amer. Math. So. 86 (98) R.J. Libera and E.J. Zlotiewi Early oeffiients of the inverse of a regular onvex funtion Pro. Amer. Math. So. 85 (98) W. Ma and D. Minda An internal geometri harateriation of strongly starlie funtions Ann. Univ. Mariae Curie-Slodowsa Set. A 5 (99) W. Ma and D. Minda A unified treatment of some speial lasses of univalent funtions Pro. Conf. on Complex Analysis Tianjin (99) M. Nunoawa and S. Owa On ertain onditions for starlieness Southeast Asian Bull. Math. 5 (00) Ch. Pommerene Univalent funtions Vandenhoe and Rupreht Göttingen J. Staniewi Some remars onerning starlie funtions Bull. Aad. Polon. Si. Ser. Si. Math. Astronom. Phys. 8 (970) J. Staniewi On a family of starlie funtions Ann. Univ. Mariae Curie-Slodowsa Set. A - (968/70) 75 8.