Others

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

Description
Coefficient estimates for certain subclasses of analytic and bi-univalent functions
Categories
Published
of 10
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  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.  [18]and Kuroki and Owa [11], 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 [7] 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 [11], 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.  ([8])  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 [11], 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.  ([13]) 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.
Search
Tags
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks