# Coefficient estimates for certain subclasses of analytic and bi-univalent functions

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Coefficient estimates for certain subclasses of analytic and bi-univalent functions
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Filomat 29:2 (2015), 351–360DOI 10.2298  /  FIL1502351S Published by Faculty of Sciences and Mathematics,University of Niˇs, SerbiaAvailable at:  http://www.pmf.ni.ac.rs/filomat Coe ﬃ cient Estimates for Certain Subclasses of Analytic andBi-univalent Functions Yong Sun a  , Yue-Ping Jiang a  , Antti Rasila b a School of Mathematics and Econometrics, Hunan University, Changsha 410082, China b Department of Mathematics and Systems Analysis, Aalto University, Aalto, P. O. Box 11100, FI-00076, Finland Abstract.  For  λ  ≥  0 and 0  ≤  α <  1  < β , we denote by K  ( λ ; α,β ) the class of normalized analytic functionssatisfying the two sided-inequality α <  ℜ   zf  ′ (  z )  f  (  z )  + λ  z 2  f  ′′ (  z )  f  (  z )   < β  (  z  ∈ U ) , where U is the open unit disk. Let  K  Σ ( λ ; α,β ) be the class of bi-univalent functions such that  f   and itsinverse  f  − 1  both belong to the class  K  ( λ ; α,β ). In this paper, we establish bounds for the coe ﬃ cients, andsolve the Fekete-Szeg˝o problem, for the class K  ( λ ; α,β ). Furthermore, we obtain upper bounds for the ﬁrsttwo Taylor-Maclaurin coe ﬃ cients of the functions in the class K  Σ ( λ ; α,β ). 1. Introduction Let A denote the class of the functions of the form:  f  (  z )  =  z + ∞  n = 2 a n  z n ,  (1.1)which are analytic in the open unit disk U =  {  z  ∈ C :  |  z |  <  1 } , and let S  be the class of functions in A whichare univalent in U .It is well known that every function  f   ∈ S of the form (1.1) has an inverse  f  − 1 , deﬁned by  f  − 1   f  (  z )   =  z  (  z  ∈ U ) , and  f    f  − 1 ( w )   =  w  | w |  <  r ;  r  ≥  14  , 2010  Mathematics Subject Classiﬁcation . Primary 30C45; Secondary 30C50. Keywords . Univalent analytic function; Bi-univalent function; Coe ﬃ cient bound.Received: 20 October 2014; Accepted: 19 January 2015Communicated by Miodrag Mateljevi´cResearchsupportedbyNationalNaturalScienceFoundationofChina(GrantNo. 11371126)andAcademyofFinlandandNationalNatural Science Foundation of China (Grant No. 269260). Email addresses:  yongsun2008 @ foxmail.com   (Yong Sun),  ypjiang731 @ 163.com   (Yue-Ping Jiang),  antti.rasila@iki.fi  (AnttiRasila)  Y. Sun, Y.-P. Jiang, A. Rasila  /   Filomat 29:2 (2015), 351–360  352where  f  − 1 ( w )  =  w − a 2 w 2 +  2 a 22  − a 3  w 3 −  5 a 22  − 5 a 2 a 3  + a 4  w 4 + ···  .  (1.2)A function  f   ∈ A  is bi-univalent in  U  if both  f   and  f  − 1 are univalent in  U . Let  Σ  denote the class of  bi-univalent functions deﬁned in the open unit disk U . Recently, the bounds of coe ﬃ cients of analytic and bi-univalent functions have been studied by many authors. We refer the reader to [2, 3, 5, 12, 14–17, 19, 20]for recent investigations in this topic.For two analytic functions  f   and    in U , we say that  f   is subordinate to    in U , and write  f   ≺    (  z  ∈ U ),if   f  (  z )  =    ω (  z )   (  z  ∈ U )for some analytic function  ω (  z ) such that ω (0)  =  0 and  | ω (  z ) |  <  1 (  z  ∈ U ) . If     is univalent in U , then the subordination  f   ≺    is equivalent to  f  (0)  =   (0) and  f  ( U )  ⊂   ( U ) . A function  f   ∈ A is said to be starlike of order  α  (0  ≤  α <  1), if it satisﬁes the condition ℜ   zf  ′ (  z )  f  (  z )   > α  (  z  ∈ U ) . We denote  S ∗ ( α ) by the class of starlike functions of order  α . Also, we denote  M (  β ) be the subclass of   A consisting of functions  f  (  z ) which satisfy the inequality ℜ   zf  ′ (  z )  f  (  z )   < β  (  z  ∈ U ) , for some  β >  1. Moreover, the subclass  S ∗ ( α,β )  ⊂ A  consists of functions, which satisfy the followinginequality α <  ℜ   zf  ′ (  z )  f  (  z )   < β  0  ≤  α <  1  < β ;  z  ∈ U  . We remark that the functions classes  M (  β ) and S ∗ ( α,β ) were ﬁrst investigated by Uralegaddi  et al.  and Kuroki and Owa , respectively.Next we consider the following two new subclasses of  A . Deﬁnition 1.1.  Let  λ ,  α  and  β  be real numbers such that  λ  ≥  0 and 0  ≤  α <  1  < β . A function  f   ∈ A  belongsto the class K  ( λ ; α,β ) if   f   satisﬁes the inequality: α <  ℜ   zf  ′ (  z )  f  (  z )  + λ  z 2  f  ′′ (  z )  f  (  z )   < β  (  z  ∈ U ) . Remark 1.2.  If we set  λ  =  0 in Deﬁnition 1.1, then it reduces to the class  S ∗ ( α,β ). It is clear that  S ∗ ( α,β )  ⊂S ∗ ( α ) and S ∗ ( α,β )  ⊂ M (  β ). Deﬁnition 1.3.  Let  λ  ≥  0 and 0  ≤  α <  1  < β , we denote by  K  Σ ( λ ; α,β ) the class of bi-univalent functionsconsisting of the functions in A such that  f   ∈ K  ( λ ; α,β ) and  f  − 1 ∈ K  ( λ ; α,β ) , where  f  − 1 is the inverse function of   f  .  Y. Sun, Y.-P. Jiang, A. Rasila  /   Filomat 29:2 (2015), 351–360  353 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 -  0.50.00.5 Figure 1: The image of  D under the function  p (  z ) for  α  =  1 / 2 and  β  =  2. Remark 1.4.  If   λ  =  0 in Deﬁnition 1.3, for simplicity, we write S ∗ Σ ( α,β ) instead of  K  Σ (0; α,β ).A classical theorem of Fekete and Szeg˝o  states that for  f   ∈ S of the form (1.1), the functional  a 3 − λ a 22  satisﬁes the inequality  a 3  − λ a 22   ≤  3 − 4 λ, λ  ≤  0 , 1 + 2 e − (2 λ ) / (1 − λ ) ,  0  ≤  λ  ≤  1 , 4 λ − 3 , λ  ≥  1 . This inequality is sharp in the sense that for each real  λ  there exists a function in S such that equality holds(see [1, 9]). Thus the determination of sharp upper bounds for the nonlinear functional  a 3  − λ a 22   for anycompact family F   of functions in A is often called the Fekete-Szeg˝o problem for F  .This paper is organized as follows. We start with coe ﬃ cient estimates for functions of the classes K  ( λ ; α,β ) and  K  Σ ( λ ; α,β ). The ﬁrst of our main results, Theorem 3.1, gives bounds of coe ﬃ cients for thethe functions of the class  K  ( λ ; α,β ). The second of our main results, Theorem 3.4, solves the Fekete-Szeg˝oproblem for the class K  ( λ ; α,β ). Finally, in Theorem 3.6, we estimate the upper bounds of initial coe ﬃ cientsof inverse functions and bi-univalent functions of the class K  Σ ( λ ; α,β ). 2. Preliminary Results In , Kuroki and Owa deﬁned an analytic function  p : U → C  by  p (  z )  =  1 + (  β − α ) i π  log  1 −  ze 2 π (1 − α ) i / (  β − α ) 1 −  z   (0  ≤  α <  1  < β ;  z  ∈ U ) ,  (2.1)and they proved that  p  maps U onto the convex domain (see Figure 1) Ω =  ω  :  α <  ℜ ( ω )  < β  . We observe that the function  p , deﬁned by (2.1), has the representation  p (  z )  =  1 + ∞  n = 1 B n  z n (  z  ∈ U ) ,  (2.2)  Y. Sun, Y.-P. Jiang, A. Rasila  /   Filomat 29:2 (2015), 351–360  354where B n  = (  β − α ) in π  1 − e 2 n π (1 − α ) i / (  β − α )   ( n  ∈ N ) .  (2.3)In order to prove our main results, we need the following lemmas. Lemma 2.1.  ()  Let p (  z )  =  1 + c 1  z + c 2  z 2 + ···  be a function with positive real part in U . Then, for any complexnumber  ν ,  c 2  −  ν c 21   ≤  2max { 1 , | 1 − 2  ν |} . The proof of the next lemma is similar to that of Lemma 1.3 in , and we omit the details. Lemma 2.2.  Let f   ∈ A and  0  ≤  α <  1  < β . Then f   ∈ K  ( λ ; α,β )  if and only if  zf  ′ (  z )  f  (  z )  + λ  z 2  f  ′′ (  z )  f  (  z )  ≺  p (  z ) (  z  ∈ U ) ,  (2.4) where p (  z )  is given by  (2.1) . Lemma 2.3.  () Letp (  z )  =  ∞ n = 1  C n  z n beanalyticandunivalentin U andsupposethatp (  z ) maps U ontoaconvexdomain. If q (  z )  =  ∞ n = 1  A n  z n is analytic in U and satisﬁes the subordination:q (  z )  ≺  p (  z ) (  z  ∈ U ) , then |  A n | ≤ | C 1 |  ( n  =  1 , 2 ,... ) . 3. Main Results We begin by presenting some coe ﬃ cient problems involving functions of the class K  ( λ ; α,β ). Theorem 3.1.  If f   ∈ K  ( λ ; α,β ) , then | a 2 | ≤ | B 1 | 2 λ + 1 and  | a n | ≤ | B 1 | ( n − 1)( n λ + 1) n − 1  k  = 2  1 +  | B 1 | ( k  − 1)( k  λ + 1)   ( n  =  3 , 4 , 5 ,... ) ,  (3.1) where | B 1 | is given by | B 1 |  = 2(  β − α ) π  sin  π (1 − α )  β − α .  (3.2) Proof.  Let us deﬁne q (  z )  =  zf  ′ (  z )  f  (  z )  + λ  z 2  f  ′′ (  z )  f  (  z ) (  z  ∈ U ) ,  (3.3)and let the function  p  be given by (2.1). Then, the subordination (2.4) can be written as follows: q (  z )  ≺  p (  z ) (  z  ∈ U ) .  (3.4)Note that the function  p  deﬁned by (2.1) is convex in U and has the form  p (  z )  =  1 + ∞  n = 1 B n  z n (  z  ∈ U ) ,  Y. Sun, Y.-P. Jiang, A. Rasila  /   Filomat 29:2 (2015), 351–360  355where  B n  is given by (2.3). If we let q (  z )  =  1 + ∞  n = 1  A n  z n (  z  ∈ U ) , then from Lemma 2.3 we see that the subordination (3.4) implies |  A n | ≤ | B 1 |  ( n  =  1 , 2 ,... ) ,  (3.5)where | B 1 | is given by (3.2).Now, (3.3) implies that  zf  ′ (  z ) + λ  z 2  f  ′′ (  z )  =  q (  z )  f  (  z ) (  z  ∈ U ) . Then, by comparing the coe ﬃ cients of   z n on the both sides, we see that a n  =  1( n − 1)( n λ + 1)  ×   A n − 1  + a 2  A n − 2  + a 3  A n − 3  + ··· + a n − 1  A 1  . A simple calculation together with the inequality (3.5) yields that | a n |  =  1( n − 1)( n λ + 1)  ×   A n − 1  + a 2  A n − 2  + a 3  A n − 3  + ··· + a n − 1  A 1  ≤  1( n − 1)( n λ + 1)  ×  |  A n − 1 | + | a 2 ||  A n − 2 | + | a 3 ||  A n − 3 | + ··· + | a n − 1 ||  A 1 |  ≤ | B 1 | ( n − 1)( n λ + 1) n − 1  k  = 1 | a k  | , where | B 1 | is given by (3.2) and | a 1 |  =  1. Hence, we have | a 2 | ≤ | B 1 | / (2 λ + 1). To prove the remaining part of the theorem, we need to show that | B 1 | ( n − 1)( n λ + 1) n − 1  k  = 1 | a k  | ≤ | B 1 | ( n − 1)( n λ + 1) n − 1  k  = 2  1 +  | B 1 | ( k  − 1)( k  λ + 1)  ,  (3.6)for  n  =  3 , 4 , 5 ,... . We use induction to prove (3.6). The case  n  =  3 is clear. Next, assume that the inequality(3.6) holds for  n  =  m . Then, a straightforward calculation gives | a m + 1 | ≤ | B 1 | m [( m + 1) λ + 1] m  k  = 1 | a k  |  =  | B 1 | m [( m + 1) λ + 1]  m − 1  k  = 1 | a k  | + | a m |  ≤ | B 1 | m [( m + 1) λ + 1] m − 1  k  = 2  1 +  | B 1 | ( k  − 1)( k  λ + 1)  +  | B 1 | m [( m + 1) λ + 1]  × | B 1 | ( m − 1)( m λ + 1) m − 1  k  = 2  1 +  | B 1 | ( k  − 1)( k  λ + 1)  =  | B 1 | m [( m + 1) λ + 1] m  k  = 2  1 +  | B 1 | ( k  − 1)( k  λ + 1)  , whichimpliesthattheinequality(3.6)holdsfor n  =  m + 1. Hence,thedesiredestimatefor | a n | ( n  =  3 , 4 , 5 ,... )follows, as asserted in (3.1). This completes the proof of Theorem 3.1.

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