Logarithmic coefficients of the inverse of univalent functions
aa r X i v : . [ m a t h . C V ] N ov LOGARITHMIC COEFFICIENTS OF THE INVERSE OF UNIVALENTFUNCTIONS
SAMINATHAN PONNUSAMY, NAVNEET LAL SHARMA, AND KARL-JOACHIM WIRTHS
Abstract.
Let S be the class of analytic and univalent functions in the unit disk | z | < f ( z ) = z + P ∞ n =2 a n z n . Let F be the inverse of the function f ∈S with the series expansion F ( w ) = f − ( w ) = w + P ∞ n =2 A n w n for | w | < /
4. The logarithmicinverse coefficients Γ n of F are defined by the formula log ( F ( w ) /w ) = 2 P ∞ n =1 Γ n ( F ) w n .In this paper, we first determine the sharp bound for the absolute value of Γ n ( F ) when f belongs to S and for all n ≥
1. This result motivates us to carry forward similar problemsfor some of its important geometric subclasses. In some cases, we have managed to solve thisquestion completely but in some other cases it is difficult to handle for n ≥
4. For example,in the case of convex functions f , we show that the logarithmic inverse coefficients Γ n ( F ) of F satisfy the inequality | Γ n ( F ) | ≤ n for n ≥ , , l ( z ) = z/ (1 − z ). Although this cannot be truefor n ≥
10, it is not clear whether this inequality could still be true for 4 ≤ n ≤ Introduction
Let A be the class of functions f analytic in the unit disk D = { z : | z | < } of the form(1.1) f ( z ) = z + ∞ X n =2 a n z n . The subclass of A consisting of all univalent functions f in D is denoted by S . The theoryof univalent functions with a strong foundation from the class S is beautiful when it isbeing considered both by geometric and analytic considerations, and in addition, logarithmicrestrictions and special exponentiation methods are often useful. During 1960’s, Milin [15]intensively investigated the impact of transferring the properties of the logarithmic coefficientsto that of the Taylor coefficients of univalent functions themselves or to its powers and thus,their role in the theory of univalent functions. The inequalities conjectured by Milin attractedmuch attention because their truth would imply the truth of the Robertson conjecture andthe Bieberbach conjecture, in addition to few others [6, 15, 19]. It is then, in 1984, Louis deBranges [3] proved these inequalities and his proof resolved the most popular problem for theclass S , namely, the statement max f ∈S | a n | = n which occurs if and only if f is a rotationof the Koebe function k ( z ) = z/ (1 − z ) . The proof which settles the Bieberbach conjecturerelied not on the coefficients { a n } of f but rather the logarithmic coefficients { γ n } of f . Here Mathematics Subject Classification.
Key words and phrases.
Univalent function, Inverse function, starlike, spirallike, close-to-convex, and con-vex functions, subordination, Inverse Logarithmic coefficients, Schwarz’ lemma .File: SamySharWirth2˙2018˙˙InvLogCoeff˙ArXivSubmitted.tex, printed: 9-11-2018, 16.41. the logarithmic coefficients γ n of f ∈ S are defined by the formulalog (cid:18) f ( z ) z (cid:19) = 2 ∞ X n =1 γ n ( f ) z n for z ∈ D . We use γ n ( f ) = γ n when there is no confusion, and remark that some authors use γ n in placeof 2 γ n .Let F be the inverse function of f ∈ S defined in a neighborhood of the origin with theTaylor series expansion(1.2) F ( w ) := f − ( w ) = w + ∞ X n =2 A n w n , where we may choose | w | < /
4, as we know from Koebe’s 1 / | A n | ≤ K n for each n, where K n = (2 n )! / ( n !( n + 1)!) and K ( w ) = w + K w + K w + · · · is the inverse of the Koebefunction. There has been a good deal of interest in determining the behavior of the inversecoefficients of f given in (1.2) when the corresponding function f is restricted to some propergeometric subclasses of S . Alternate proofs of the inequality (1.3) have been given by severalauthors but a simpler proof was given by Yang [26]. As with f , the logarithmic coefficientsΓ n , n ∈ N , of F are defined by the equation(1.4) log (cid:18) F ( w ) w (cid:19) = 2 ∞ X n =1 Γ n ( F ) w n for | w | < / . We have a natural and fundamental question.
Problem 1.
Suppose that f ∈ S or of its subclasses and F is the corresponding inverse of f with the form (1.2) . If Γ n ( F ) denotes the logarithmic inverse coefficients of F , is it possibleto determine the sharp bound for the absolute value of Γ n ( F ) ? The main aim of this article is to deal with this problem for S and some of its importantgeometric subclasses. The article is organized as follows. In Section 2, we solve this problemcompletely for the family S which motivates the rest of the investigation. In Section 3, weintroduce the classes for which we study this problem, and present solutions to this problemin several subsections with necessary background materials.2. Logarithmic inverse coefficients for the class S Before we continue to study Problem 1 in detail, it is appropriate to deal with the class S which motivates us to consider further investigation. Let S ∗ denote the class of starlikefunctions f (i.e f ( D ) is a domain starlike with respect to the origin) in S .Recall that, for f ∈ S and λ >
0, the function ( z/f ( z )) λ is analytic in D and has the powerseries expansion of the form(2.1) g ( z ) = (cid:18) zf ( z ) (cid:19) λ = 1 + ∞ X n =1 b n ( λ, f ) z n . nverse Logarithmic coefficients problems 3 Throughout we use this representation. For the logarithmic inverse coefficients Γ n of F asgiven by (1.4), the following theorem, whose proof is elegant, is fundamental in this line ofdiscussion. Theorem 1.
Let f ∈ S (or S ∗ ) and F be the inverse function of f and have the form (1 . .Then for n ∈ N , the logarithmic inverse coefficients Γ n of F satisfy the sharp inequality | Γ n ( F ) | ≤ n (cid:18) nn (cid:19) . Equality is attained if and only if f is the Koebe function or one of its rotations.Proof. The idea of proof of here is well-known and Lagrange’s series have a similar idea ofthe proof. We consider ddw (cid:18) w log (cid:18) F ( w ) w (cid:19)(cid:19) = wF ′ ( w ) F ( w ) − ∞ X n =1 n Γ n ( F ) w n . Using the Cauchy integral formula and the relation (2.1), it is easy to obtain the followingidentity for each n ∈ N , 2 n Γ n ( F ) = 12 πi Z C F ′ ( w ) F ( w ) w n dw = 12 πi Z F ( C ) (cid:18) zf ( z ) (cid:19) n z n +1 dz = b n ( n, f ) , (2.2)where C is a Jordan curve surrounding the origin counterclockwise in f ( D ). Concerning thisidentity, see [24, Theorem 3]. With the use of L¨owner’s method [12], it has been proved in[24] that | b n ( λ, f ) | ≤ | b n ( λ, k ) | = (cid:18) λn (cid:19) for λ > , where k equals the Koebe function k ( z ) = z/ (1 − z ) . Hence,2 n | Γ n ( F ) | ≤ b n ( n, k ) , n ∈ N . Likewise, it was proved in [24] that equality occurs if and only if f equals k or one of itsrotations. Since (cid:18) zk ( z ) (cid:19) n = (1 − z ) n = n X j =0 ( − j (cid:18) nj (cid:19) z j , the binomial theorem implies our assertion. (cid:3) Logarithmic inverse coefficients for the preliminary classes
Basic preliminary classes of study.
Let B denote the class of all analytic functions φ in D which satisfy the condition | φ ( z ) | < z ∈ D . Functions in B := { φ ∈ B : φ (0) = 0 } are called Schwarz functions. Let f and g be two analytic functions in D . We say that f is subordinate to g , written as f ≺ g , if there exists a function φ ∈ B such that f ( z ) = g ( φ ( z ))for z ∈ D . In particular, if g is univalent in D , then f ≺ g is equivalent to f ( D ) ⊂ g ( D ) and f (0) = g (0). S. Ponnusamy, N. L. Sharma, and K.-J. Wirths
The following subclasses of S have been studied extensively in the literature. See [9] and[17, 21, 22] and the references therein.(1) The class S ∗ ( A, B ) is defined by S ∗ ( A, B ) := (cid:26) f ∈ A : zf ′ ( z ) f ( z ) ≺ Az Bz for z ∈ D (cid:27) , where A ∈ C and − ≤ B ≤
0, and this class has been studied extensively in theliterature. For 0 ≤ β < S ∗ ( β ) := S ∗ (1 − β, −
1) is the class of starlike functionsof order β . In particular, for B = − A = e iα ( e iα − β cos α ), the class S ∗ ( A, B )reduces to the class S α ( β ) of spiral-like functions of order β defined by S α ( β ) := (cid:26) f ∈ A : Re (cid:18) e − iα zf ′ ( z ) f ( z ) (cid:19) > β cos α, z ∈ D (cid:27) , where β ∈ [0 ,
1) and α ∈ ( − π/ , π/ S α ( β ) is univalent in D (see[11]). Clearly, S α ( β ) ⊂ S α (0) ⊂ S whenever 0 ≤ β <
1. Functions in S α (0) are called α -spirallike , but they do not necessarily belong to the starlike family S ∗ := S ∗ (1 , − G ( c ) is defined by G ( c ) := (cid:26) f ∈ A : Re (cid:18) zf ′′ ( z ) f ′ ( z ) (cid:19) < c , z ∈ D (cid:27) , where c ∈ (0 , G (1) =: G . It is known that G ⊂ S ∗ and thus, functions in G ( c ) are starlike. This class has been studied extensively in the recent past, see forinstance [16, 18] and the references therein.(3) The class U ( λ ) is defined by U ( λ ) := ( f ∈ A : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ′ ( z ) (cid:18) zf ( z ) (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < λ, z ∈ D ) , where 0 < λ ≤
1. Set U := U (1), and observe that U ( S . See [1, 2]. Many propertiesof U ( λ ) and its various generalizations have been investigated in the literature, we referfor example [17, 22] and the references therein.(4) The class F ( α ) is defined by F ( α ) := (cid:26) f ∈ A : Re (cid:18) zf ′′ ( z ) f ′ ( z ) (cid:19) > α, z ∈ D (cid:27) for α ∈ [ − / , . In particular, we let F (0) =: C . Functions in C known to be convexand univalent in D (i.e f ( D ) is a convex domain). For α ∈ [0 , F ( α )are convex functions of order α in D , and it is usually denoted by C ( α ). The functionsin F ( − /
2) (and hence in F ( α ) for α ∈ [ − / , D .3.2. Logarithmic inverse coefficients for f ∈ S ∗ ( A, B ) . Throughout in the sequel, let I k ( n ) denote the semi-closed intervals (cid:2) kn , k +1 n (cid:1) for k = 0 , , . . . , n − n ∈ N . Theorem 2.
Let f ∈ S ∗ ( A, B ) , δ = (1 − A ) / (1 − B ) with − ≤ B < A ≤ , and k A,B ; n ( z ) = z (1 + Bz n ) ( A − B ) /nB . Then for n ∈ N , the logarithmic inverse coefficients Γ n of F satisfy thefollowing inequalities: nverse Logarithmic coefficients problems 5 (1) when n ∈ N and n (1 − δ ) / ∈ N , we have (3.1) | Γ n ( F ) | ≤ n n − Y j =0 n ( A − B ) + Bj j for δ ∈ I ( n ) = [0 , /n ) . (2) when n ∈ N and δ ∈ I k ( n ) , k = 1 , , . . . , n − , we have (3.2) | Γ n ( F ) | ≤ n − k n n − k − Y j =0 n ( A − B ) + Bj j . (3) when n ∈ N and n (1 − δ ) ∈ N , (3 . holds for δ ∈ I ( n ) , and (3 . holds for δ ∈ I k ( n ) , k = 2 , , . . . , n − . (4) for δ ∈ I n − ( n ) , we have (3.3) | Γ n ( F ) | ≤ A − B n , n ∈ N . The inequalities (3.1) and (3.3) are sharp for the functions k A,B ;1 ( z ) and k A,B ; n ( z ) , respec-tively.Proof. Suppose f ∈ S ∗ ( A, B ). From the relation (2.2), we have2 n Γ n ( F ) = b n ( n, f ) , n ∈ N , where b n ( n, f ) is defined by (2.1). In order to compute | Γ n ( F ) | , we shall have to estimate | b n ( n, f ) | and for this, we use [7, Theorem 2.7]. It is worth to remark that one could usethe analysis used in [13]. First, take λ = n ∈ N , we note that the inequality (2.9) in [7] isapplicable for k = 0 (in case of n (1 − δ ) ∈ N , the inequality (2.9) in [7] is also applicable butonly for k = 1). Therefore for δ ∈ I = [0 , /n ), the inequality (2.9) in [7] yields | Γ n ( F ) | = 12 n | b n ( n, f ) | ≤ n n − Y j =0 n ( A − B ) + Bj j for n ∈ N , which is precisely the inequality (3.1). The equality holds for k A,B ;1 ( z ) = z (1 + Bz ) ( A − B ) /B .We have (cid:18) zk A,B ;1 ( z ) (cid:19) n = (1 + Bz ) − ξ = ∞ X m =0 ( − m ( ξ ) m B m ( m )! z m , where ξ = ( A − B ) n/B and ( a ) m = Γ( a + m ) / Γ( a ) denotes the Pochhammer symbol. Similarly,for δ ∈ I k ( n ) , k = 1 , , . . . , n −
1, the inequality (2.10) in [7] gives | Γ n ( F ) | = 12 n | b n ( n, f ) | ≤ n − k n n − k − Y j =0 n ( A − B ) + Bj j . This gives (3.2). Finally, for δ ∈ I n − ( n ), the inequality (2.11) in [7] yields | Γ n ( F ) | = 12 n | b n ( n, f ) | = A − B n , n ∈ N , which gives (3.3). It is easily verified that equality holds in (3.3) as the function k A,B ; n ( z ) = z (1 + Bz n ) ( A − B ) /nB demonstrates. This completes the proof of Theorem 2. (cid:3) S. Ponnusamy, N. L. Sharma, and K.-J. Wirths
Theorem 2 for the case A = 1 − β and B = − Corollary 3.
Let f ∈ S ∗ ( β ) for some β ∈ [0 , , and k β ; n ( z ) = z/ (1 − z n ) − β ) /n . Then thelogarithmic inverse coefficients Γ n of F satisfy the inequalities: (1) for n ∈ N and β ∈ [0 , /n ) , we have (3.4) | Γ n ( F ) | ≤ n n − Y j =0 n (1 − β ) − j j , (2) for n ∈ N and β ∈ I k ( n ) , k = 1 , , . . . , n − , we have | Γ n ( F ) | ≤ n − k n n − k +1 Y j =0 n (1 − β ) − j j (3) for β ∈ I n − ( n ) , we have (3.5) | Γ n ( F ) | ≤ − βn , n ∈ N . The inequalities (3.4) and (3.5) are sharp for the functions k β ;1 ( z ) and k β ; n ( z ) , respectively. We remark that when A = 1 and B = − β = 0 in Corollary 3, weobtain Theorem 1 for f ∈ S ∗ . Moreover, we can generalize Corollary 3 for the class S α ( β ) ofspiral-like functions of order β .3.3. Logarithmic inverse coefficients for S α ( β ) . In [13], the authors proved the followingtheorem (which we state in our form).
Theorem A.
Suppose f ( z ) = z + P ∞ n = p +1 a n z n ∈ S α ( β ) ( | α | < π/ , ≤ β < , and forintegral t ≥ , let (cid:18) zf ( z ) (cid:19) t = 1 + ∞ X k = p b ( p ) k ( t, f ) z k , < | z | < . Then | b ( p ) k ( t, f ) | ≤ m pk m − Y j =0 (cid:18) | (2 t/p )(1 − β ) cos α e − iα − j | j + 1 (cid:19) for − mp ≤ k ≤ ( m + 1) p − , where m = 1 , . . . , M + 1 , and M = [ t (1 − β ) /p ] . Here [ x ] denotes the largest integer notexceeding x . Setting p = 1 gives that if f ( z ) = z + P ∞ n =2 a n z n ∈ S α ( β ) and b (1) k ( t, f ) =: b k ( t, f ), then wehave(3.6) | b k ( t, f ) | ≤ k − Y j =0 (cid:18) | t (1 − β ) cos α e − iα − j | j + 1 (cid:19) where k = 1 , . . . , M + 1, and M = [ t (1 − β )]. Moreover,2 n Γ n ( F ) = b n ( n, f ) , n ∈ N , nverse Logarithmic coefficients problems 7 and thus, by taking t = n ∈ N in (3.6), we obtain | Γ n ( F ) | ≤ n k − Y j =0 (cid:18) | n (1 − β ) cos α e − iα − j | j + 1 (cid:19) where k = 1 , . . . , [ n (1 − β )] + 1.This is the basic and we organize it in the following form. We use results from [7] andTheorem 2 to prove the following. Theorem 4.
Let f ∈ S α ( β ) for some β ∈ [0 , and α ∈ ( − π/ , π/ . Then the logarithmicinverse coefficients of F satisfy the inequalities: (1) for n ∈ N and β ∈ I ( n ) = [0 , /n ) , we have (3.7) | Γ n ( F ) | ≤ n n − Y j =0 | n (1 − β ) e − iα cos α − j | j (2) for n ∈ N and β ∈ I k ( n ) , k = 1 , , . . . , n − , we have (3.8) | Γ n ( F ) | ≤ n − k n n − k − Y j =0 | n (1 − β ) e − iα cos α − j | j . (3) for β ∈ I n − ( n ) , we have (3.9) | Γ n ( F ) | ≤ (1 − α ) cos βn . The estimates (3 . and (3 . are sharp for f α,β ; n ( z ) = z/ (1 − z n ) γ/n , γ = 2(1 − β ) cos α and f α,β ;1 ( z ) , respectively.Proof. Suppose f ∈ S α ( β ). From the relation (2.2), we have2 n Γ n ( F ) = b n ( n, f ) , n ∈ N , where b n ( n, f ) is defined by (2.1). In order to find | Γ n ( F ) | , we need to estimate | b n ( n, f ) | with the help of [25, Theorem 4]. First, take λ = n ∈ N , we note that the inequality (47) in[25] is applicable only for k = 0. Therefore for β ∈ [0 , /n ), the inequality (47) in [25] yields | Γ n ( F ) | = 12 n | b n ( n, f ) | ≤ n n − Y j =0 | n (1 − β ) e − iα cos α − j | j for n ∈ N , which is precisely the inequality (3.7). The equality holds for f α,β ( z ) = z/ (1 − z ) γ , γ =2(1 − β ) cos α . We note that (cid:18) zf α,β ( z ) (cid:19) n = (1 − z ) − θ = ∞ X m =0 ( θ ) m ( m )! z m . where θ = − nγ . Similarly, for β ∈ I k ( n ) , k = 1 , , . . . , n −
1, the inequality (48) in [25] yields(3.8). Finally, for β ∈ I n − ( n ), we note that the inequality (49) in [25] gives | Γ n ( F ) | = 12 n | b n ( n, f ) | ≤ (1 − α ) cos βn , n ∈ N , which establishes (3.9). It is easily verified that equality holds in (3.9) for the function f α,β ; n ( z ) = z/ (1 − z n ) γ/n . This completes the proof of Theorem 4. (cid:3) S. Ponnusamy, N. L. Sharma, and K.-J. Wirths
Logarithmic inverse coefficients for G ( c ) .Lemma 5. Let f ∈ G ( c ) for some c ∈ (0 , and for each fixed λ > , the Taylor coefficients b m ( λ, f ) be given by (2 . . Then (1) for λ ∈ (0 , , we have (3.10) | b m ( λ, f ) | ≤ λcm (1 + c ) for m = 1 , , . . . ;(2) for λ > , we have (3.11) | b m ( λ, f ) | ≤ c ) m m − Y j =0 λc + j j for m = 1 , , . . . , [ λ ] + 1; and (3.12) | b m ( λ, f ) | ≤ [ λ ] m (1 + c ) [ λ ] [ λ ] − Y j =0 λc + j j for m = [ λ ] + 2 , [ λ ] + 3 , . . . ; The estimates (3 . and (3 . are sharp for the function f ′ c,m ( z ) = (1 − z m ) cm and f ′ c ;1 ( z ) = f ′ c ( z ) , respectively.Proof. Suppose that f ∈ G ( c ). Then we have (see [20]) zf ′ ( z ) f ( z ) − ≺ (1 + c )(1 − z )1 + c − z − − cz c − z . As g ( z ) = (cid:18) zf ( z ) (cid:19) λ = 1 + ∞ X n =1 b n ( λ, f ) z n , by the definition of subordination, there exists an analytic function ϕ ∈ B such that zg ′ ( z ) − λg ( z ) = zf ′ ( z ) f ( z ) − − cϕ ( z )1 + c − ϕ ( z )or equivalently (1 + c ) zg ′ ( z ) = ϕ ( z )( λcg ( z ) + zg ′ ( z )) . As with the standard procedure, we may write this in series form as(1 + c ) m X k =1 kb k ( λ, f ) z k + ∞ X k = m +1 d k ( λ, f ) z k = ϕ ( z ) λc + m − X k =1 ( λc + k ) b k ( λ, f ) z k ! , the second sum on the left-hand side being convergent in D . By Clunie’s method [4, 5] (seealso Parseval-Gutzmer formula) together with | ϕ ( z ) | < c ) m | b m ( λ, f ) | ≤ λ c + m − X k =1 [( λc + k ) − k (1 + c ) ] | b k ( λ, f ) | . Since ( λc + k ) − k (1 + c ) = c ( λ − k )[ c ( λ + k ) + 2 k ], the sign of each term inside thesummation symbol on the right-hand side of (3.13) depends on the expression ( λ − k ) for k = 1 , , . . . , m − nverse Logarithmic coefficients problems 9 Case I: If λ ∈ (0 , λ − k ≤ k = 1 , , . . . , m −
1. Then from (3.13), we find that | b m ( λ, f ) | ≤ λcm (1 + c ) for m = 1 , , . . . , which establishes the inequality (3.10). Case II: If λ > , then λ − k > k = 1 , , . . . , [ λ ], and λ − k ≤ k = [ λ ] + 1 , [ λ ] + 2 , . . . .Therefore from (3.13), for m = 1 , , . . . , [ λ ] + 1, we obtain(3.14) m | b m ( λ, f ) | ≤ c ) λ c + m − X k =1 [( λc + k ) − k (1 + c ) ] | b k ( λ, f ) | ! . Now we use the principle of mathematical induction on m . For m = 1, it follows from (3.14)that | b ( λ, f ) | ≤ λc/ (1 + c ). This gives the estimate (3.11) for m = 1. For m = 2 , . . . , [ λ ], wenow assume that(3.15) | b m ( λ, f ) | ≤ c ) m m − Y j =0 λc + j j holds. Then, using (3.14), (3.15) and simplifying, it follows that m | b m ( λ, f ) | ≤ c ) " λ c + m − X k =1 (cid:0) ( λc + k ) − k (1 + c ) (cid:1) c ) k k − Y j =0 (cid:18) λc + j j (cid:19) = 1(1 + c ) " λ c + m − X k =1 ( A k +1 − A k ) , A k = k (1 + c ) k − k − Y j =0 λc + j j , = 1(1 + c ) A m . Hence, for m = 1 , , . . . , [ λ ] + 1, we have | b m ( λ, f ) | ≤ c ) m m − Y j =0 λc + j j . This establishes the inequality (3.11).
Case III:
Now, we will prove the inequality (3.12). Recall that if λ > , then λ − k ≤ k = [ λ ] + 1 , [ λ ] + 2 , . . . . From (3.13), for m = [ λ ] + 2 , [ λ ] + 3 , . . . , we get(3.16) m | b m ( λ, f ) | ≤ c ) λ c + [ λ ] − X k =1 [( λc + k ) − k (1 + c ) ] | b k ( λ, f ) | . Using (3.16) and mathematical induction hypothesis (3.15), we get as before m | b n ( λ, f ) | ≤ c ) λ c + [ λ ] − X k =1 (cid:0) ( λc + k ) − k (1 + c ) (cid:1) c ) k k − Y j =0 (cid:18) λc + j j (cid:19) = 1(1 + c ) λ ] (([ λ ] − λ ] − Y j =0 ( λc + j ) . Hence, | b m ( λ, f ) | ≤ [ λ ] m (1 + c ) [ λ ] [ λ ] − Y j =0 λc + j j for m = [ λ ] + 2 , [ λ ] + 3 , . . . . This establishes the inequality (3.12). (cid:3)
If we take c = 1 in Lemma 5, we get the following result. Corollary 6.
Let f ∈ G (1) and for each fixed λ > , let the Taylor coefficients b m ( λ, f ) begiven by (2 . . Then (1) for λ ∈ (0 , , we have (3.17) | b m ( λ, f ) | ≤ λ m for m = 1 , , . . . ;(2) for λ > , we have (3.18) | b m ( λ, f ) | ≤ m m − Y j =0 λ + j j for m = 1 , , . . . , [ λ ] + 1; and | b m ( λ, f ) | ≤ [ λ ] m [ λ ] [ λ ] − Y j =0 λ + j j for m = [ λ ] + 2 , . . . . The estimate (3 . is sharp for f ′ ,m and the estimate (3 . is sharp for f ( z ) = z − z / . Now we are ready to state our next main result.
Theorem 7.
Let f ∈ G ( c ) for some c ∈ (0 , . Then the logarithmic inverse coefficients Γ n of F satisfy the inequality | Γ n ( F ) | ≤ n (1 + c ) n n − Y j =0 nc + j (1 + j ) for n ∈ N . The result is best possible for the function f ′ c ( z ) = (1 − z ) c .Proof. Suppose f ∈ G ( c ). From the relation (2.2), we have2 n Γ n ( F ) = b n ( n, f ) , n ∈ N , where b n ( n, f ) is defined by (2.1). In order to find | Γ n ( F ) | , we shall estimate | b n ( n, f ) | usingLemma 5. For λ = n ∈ N , we note that the inequalities (3.10) and (3.11) are applicable.Therefore, the inequalities (3.10) and (3.11) yield | Γ n ( F ) | = 12 n | b n ( n, f ) | for n ∈ N . nverse Logarithmic coefficients problems 11 The desired conclusion follows. (cid:3)
Corollary 8.
Let f ∈ G (1) . Then | Γ n ( F ) | ≤ (2 n − n !) n +1 for n ∈ N . The result is best possible for the function f ( z ) = z − z / . Logarithmic inverse coefficients for U ( λ ) . Now, we will discuss the logarithmicinverse coefficients Γ n for the class U ( λ ). It is a simple exercise to see that f ∈ U ( λ ) if andonly if(3.19) f ( z ) = z − a z + λz R z ω ( t ) dt , where 2 a = f ′′ (0), ω is analytic and | ω ( z ) | ≤ | z | <
1. Moreover, we also see from(3.19) that f ′ ( z ) (cid:18) zf ( z ) (cid:19) − − λz ω ( z ) , where λω (0) = − ( a − a ). In [22], the authors proved that if ω (0) = a ∈ D and v ( x ) = Z x + t xt dt = 1 x − − x x log(1 + x ) , x ∈ [0 , , where v (0) = lim x → + v ( x ) = 1 /
2, then we have the sharp inequality(3.20) | a | ≤ λv ( | a | ) . Theorem 9.
Let f ∈ U ( λ ) for < λ ≤ . Then the logarithmic inverse coefficients Γ n of F satisfy the inequality | Γ ( F ) | ≤
12 [1 + λv ( | a | )] and | Γ ( F ) | ≤ (cid:2) (1 + λv ( | a | )) + 2 λ | a | (cid:3) . Equality is achieved in both inequalities for the function (3.21) f ( z ) = z − (1 + λv ( a )) z + λz R z t + a at dt , where a ∈ (0 , .Proof. Suppose f ∈ U ( λ ). Then from (3 . zf ( z ) = 1 − a z + λaz + o ( z ) , for z → . From (2.2), we know that 2 n Γ n ( F ) = b n ( n, f ) for n ∈ N , and from the last relation it followseasily that b (1 , f ) = − a and b (2 , f ) = 3 a − a = a + 2 λa. Hence, by using (3 . | Γ ( F ) | = | a | ≤ λv ( | a | ) , and 4 | Γ ( F ) | = | a + 2 λa | ≤ (1 + λv ( | a | )) + 2 λ | a | . Equality case is easy to obtain from (3.21). (cid:3)
Logarithmic inverse coefficients for F ( α ) . We see from the definition of F ( α ) thatif f ∈ F ( α ), then1 + zf ′′ ( z ) f ′ ( z ) ≺ − α ) z − z , i.e. zf ′′ ( z ) f ′ ( z ) ≺ − α ) z − z . By the definition of subordination, we get zf ′′ ( z ) f ′ ( z ) = 2(1 − α ) ϕ ( z )1 − ϕ ( z ) , i.e. zf ′′ ( z )(1 − ϕ ( z )) = 2(1 − α ) f ′ ( z ) ϕ ( z ) , where ϕ ∈ B . Using the Taylor expansion ϕ ( z ) = P ∞ k =1 c k z k and of f ( z ) given by (1.1), wecan write the above relation in the series representation a z + (3 a − a c ) z + (6 a − a c − a c ) z + . . . = (1 − α ) (cid:2) c z + (2 a c + c ) z + (3 a c + 2 a c + c ) z + . . . (cid:3) and the sharp inequality | c n | ≤ − | c | holds for n ≥
2. Now, we compare the coefficients of z n for n = 2 , , a = (1 − α ) c a = (1 − α )((3 − α ) c + c )6 a = (1 − α )((2 − α )(3 − α ) c + (5 − α ) c c + c )In view of the relation (1.2), we have f ( F ( w )) = w, F (0) = 0 = f (0) and F ′ (0) = 1 = f ′ (0) , where z = F ( w ). Differentiating this we find that f ′ ( z ) F ′ ( w ) = 1, and further differentiationgives f ′′ ( z )( F ′ ( w )) + f ′ ( z ) F ′′ ( w ) = 0 ,f ′′′ ( z )( F ′ ( w )) + 3 f ′′ ( z ) F ′ ( w ) F ′′ ( w ) + f ′ ( z ) F ′′′ ( w ) = 0 ,f ( iv ) ( z )( F ′ ( w )) + 6 f ′′′ ( z )( F ′ ( w )) F ′′ ( w ) + f ′′ ( z )[3( F ′′ ( w )) + 3 F ′ ( z ) F ′′′ ( w ) + F ′′′ ( w )]+ f ′ ( z ) F ( iv ) ( w ) = 0 . Setting z = 0 and w = 0, we obtain that(3.23) A = − a A = − a + 2 a A = − a + 5 a a − a . Next, we simplify (1.4) and write in the series form A w + A w + A w + · · · −
12 [ A w + A w + · · · ] + 13 [ A w + · · · ] + · · · = 2 ∞ X n =1 Γ n ( F ) w n Now, we compare the coefficients of w n for n = 1 , , ( F ) = A ( F ) = A − A ( F ) = A − A A + 13 A . nverse Logarithmic coefficients problems 13 From the formulas (3.24) and (3.23), we obtain(3.25) ( F ) = − a ( F ) | = − a + 3 a ( F ) = − a + 12 a a − a . Finally, the formulas (3.22) and (3.25) together yield(3.26) ( F ) = − (1 − α ) c ( F ) | = 1 − α − c + (3 − α ) c )6Γ ( F ) = 1 − α − c + (3 − α ) c c − (3 α − α − c ) . These equations imply the following sharp bounds for the logarithmic inverse coefficients.The first equation in (3.26) gives
Theorem 10.
Let f ∈ F ( α ) for some α ∈ [ − / , . Then | Γ ( F ) | ≤ − α . Equality is attained if and only if f ′ ( z ) = (1 − z ) − − α ) or a rotation of this function. The second and the third relations in (3.26) give
Theorem 11.
Let f ∈ F ( α ) for some α ∈ [ − / , . Then (a) If α ∈ [ − / , / , then | Γ ( F ) | ≤ (1 − α )(3 − α )12 . Equality is attained in each case if and only if f ′ ( z ) = (1 − z ) − − α ) or a rotation ofthis function. (b) If α ∈ (1 / , , then | Γ ( F ) | ≤ − α . Equality is attained in each case if and only if f ′ ( z ) = (1 − z ) − (1 − α ) or a rotation ofthis function.Proof. Using the sharp inequality | c | ≤ − | c | , we see that the above expression for Γ ( F )implies 4 | Γ ( F ) | ≤ − α | c | ( | − α | − | c | = 1 and in the secondcase for | c | = 0. The extremal functions are calculated using ϕ ( z ) = z in the first case and ϕ ( z ) = z in the second case.Concerning these inequalities compare [14]. (cid:3) In their paper [23], Prokhorov and Szynal calculated the maximum of the expression | c + µc c + υc | for fixed ( µ, υ ) ∈ R , where ϕ varies in the set of Schwarz functions. It is obvious that thisresult can be used to get the maximum of | Γ ( F ) | for any α ∈ [ − / , Lemma 12. [23, Lemma 2]
Let ϕ ( z ) = P ∞ k =1 c k z k ∈ B be a Schwarz function and Ψ( ϕ ) = | c + µc c + υc | . Then we have the following sharp estimates: (a) Ψ( ϕ ) ≤ if ( µ, υ ) ∈ D ∪ D , where D = (cid:26) ( µ, υ ) ∈ R : | µ | ≤ , − ≤ υ ≤ (cid:27) , and D = (cid:26) ( µ, υ ) ∈ R : 12 ≤ | µ | ≤ ,
427 ( | µ | + 1) − ( | µ | + 1) ≤ υ ≤ (cid:27) . (b) Ψ( ϕ ) ≤ | υ | if ( µ, υ ) ∈ D ∪ D , where D = (cid:26) ( µ, υ ) ∈ R : 2 ≤ | µ | ≤ , υ ≥
112 ( µ + 8) (cid:27) , and D = (cid:26) ( µ, υ ) ∈ R : | µ | ≥ , υ ≥
23 ( | µ | − (cid:27) . It is a lengthy, but straightforward verification that(3.27) ( µ, υ ) ∈ D , if α ∈ (cid:2) , (cid:3) , ( µ, υ ) ∈ D , if α ∈ (cid:2) . , (cid:3) , ( µ, υ ) ∈ D , if α ∈ (cid:2) − , (cid:3) , ( µ, υ ) ∈ D , if α ∈ (cid:2) − , − (cid:3) which help to prove the next result. Theorem 13.
Let f ∈ F ( α ) for α ∈ [ − , . Then | Γ ( F ) | ≤ − α , α ∈ (cid:20) . , (cid:21) . Equality is attained if f ′ ( z ) = (1 − z ) − − α )3 or a rotation of this function. Also, | Γ ( F ) | ≤ (1 − α )(3 α − α − , α ∈ (cid:20) − , (cid:21) . Equality is attained if f ′ ( z ) = (1 − z ) − − α ) or a rotation of this function.Proof. We see that from the expression (3.26) for Γ ( F ) implies6 | Γ ( F ) | = 1 − α (cid:12)(cid:12)(cid:12) c − (3 − α ) c c + (3 α − α − c (cid:12)(cid:12)(cid:12) =: 1 − α | I | , nverse Logarithmic coefficients problems 15 where I = c + µ c c + υ c , µ = 5 α − υ = (3 α − α − . Our aim is to get a sharp bound for | I | . Lemma 12(a) and (3.27) give | I | ≤ D ∪ D and the desired inequality follows.Using the second part of Lemma 12 and (3.27), we find that | I | ≤ | υ | = (3 α − α −
1) for D ∪ D . This completes the proof of Theorem 13. (cid:3)
If we take α = 0 and α = − / Corollary 14.
Let f ∈ C . Then | Γ n ( F ) | ≤ n for n = 1 , , . The estimates are sharp for the function l ( z ) = z/ (1 − z ) . Corollary 15. If f ∈ F ( − / , then we have the sharp inequalities | Γ ( F ) | ≤ , | Γ ( F ) | ≤ , and | Γ ( F ) | ≤ The estimates are sharp for the function f ( z ) = z − z / − z ) . Concluding Remarks
From Theorem 9, we see that logarithmic inverse coefficients for the family U ( λ ) for theremaining coefficients Γ n for n ≥ f ∈ C , | Γ n ( F ) | ≤ n cannot be valid for n ≥
10, although this is true for n = 1 , , n ≥
10, then the third Lebedev-Milin inequality (see the book by S. Gong[8, p. 80]) would imply that the moduli of the coefficients of the inverses of convex functionsare all less than 1. But this is clearly wrong at least for n ≥
10 (see Kirwan and Schober[10]). On the other hand, it is natural to ask whether the last inequality is true for othervalues of n , namely, for 4 ≤ n ≤
9. Finally, Corollary 15 shows that analog problem for theclass F ( − /
2) is also open for n ≥ . Acknowledgements.
This work was completed while the second author was at IIT Madrasfor a short period during July-August, 2018. The work of the first author is supported byMathematical Research Impact Centric Support of Department of Science and Technology(DST), India (MTR/2017/000367). The second author thanks Science and Engineering Re-search Board, DST, India, for its support by SERB National Post-Doctoral Fellowship (GrantNo. PDF/2016/001274).
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E-mail address : [email protected] N.L. Sharma, S. Ponnusamy, Department of Mathematics, Indian Institute of TechnologyMadras, Chennai-600 036, India.
E-mail address : [email protected] K.-J. Wirths, Institut f¨ur Analysis und Algebra, TU Braunschweig, 38106 Braunschweig,Germany.
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