Certain Estimates of Normalized Analytic Functions
aa r X i v : . [ m a t h . C V ] J u l CERTAIN ESTIMATES OF NORMALIZED ANALYTIC FUNCTIONS
SWATI ANAND, NAVEEN KUMAR JAIN, AND SUSHIL KUMAR
Abstract.
Let φ be a normalized convex function defined on open unit disk D . For aunified class of normalized analytic functions which satisfy the second order differential sub-ordination f ′ ( z ) + αzf ′′ ( z ) ≺ φ ( z ) for all z ∈ D , we investigate the distortion theorem andgrowth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficientand the second Hankel determinant involving the inverse coefficients are examined. Introduction
The class of all normalized analytic functions(1.1) f ( z ) = z + a z + a z + · · · in the unit disk D := { z ∈ C : | z | < } is denoted by A . Denote by S , the subclass of A consisting of univalent functions in D . Let P be the class of analytic functions p definedon D , normalized by the condition p (0) = 1 and satisfying Re( p ( z )) >
0. Let f and g beanalytic in D . Then f is subordinate to g , denoted by f ≺ g , if there exists an analyticfunction w with w (0) = 0 and | w ( z ) | < z ∈ D such that f ( z ) = g ( w ( z )). In particular,if g is univalent in D then f is subordinate to g when f (0) = g (0) and f ( D ) ⊆ g ( D ). Inthis paper we shall assume that φ is an analytic function with positive real part in D andnormalized by the conditions φ (0) = 1 and φ ′ (0) >
0. It is noted that φ ( D ) is convex. Thefunction φ is symmetric with respect to the real axis and it maps D onto a region starlikewith respect to φ (0) = 1. The Taylor series representation of the function φ is given by(1.2) φ ( z ) = 1 + B z + B z + B z + · · · , where B >
0. For such φ , Ma and Minda [21] studied the unified subclasses S ∗ ( φ ) and C ( φ ) of starlike and convex functions respectively, analytically defined as S ∗ ( φ ) = (cid:26) f ∈ A : zf ′ ( z ) f ( z ) ≺ φ ( z ) (cid:27) and C ( φ ) = (cid:26) f ∈ A : 1 + zf ′′ ( z ) f ′ ( z ) ≺ φ ( z ) (cid:27) . The authors investigated the growth, distortion and coefficient estimates for these classes.In particular, the class S ∗ ( φ ) reduces to some well known subclasses of starlike fuctions.For example, when − ≤ B < A ≤ S ∗ [ A, B ] is the class of Janowski starlike functionsintroduced by Janowski [11]. For 0 ≤ α < S ∗ [1 − α, −
1] = S ∗ α is the class of starlikefunctions of order α , introduced by Robertson [30]. The class SL = S ∗ ( √ z ), introducedby Sok´o l and Stankiewicz [35], consists of functions f ∈ A such that zf ′ ( z ) /f ( z ) lies in theregion Ω L := { w ∈ C : | w − | < } , that is, the right-half of the lemniscate of Bernoulli.Later, Mendiratta et al. [22] introduced the class S ∗ e := S ∗ ( e z ) consists of functions f ∈ A satisfying the condition | log( zf ′ ( z ) /f ′ ( z )) | <
1. In 2011, Ali et al. [3](see also [10]) studied
Mathematics Subject Classification.
Key words and phrases.
Differential subordination; Growth Theorem; Distortion Theorem; LogarithmicCoefficient; Inverse Coefficient; Hankel determinant. the class of all those functions f ∈ A which satisfy the following third order differentialequation f ( z ) + αf ′ ( z ) + γz f ′′ ( z ) = g ( z ) , where the function g is subordinate to a convex function h . In [3], the best dominant onall solutions of the differential equation in terms of double integral were obtained. Somecertain variations of the class R ( α, h ) = { f ∈ A : f ′ ( z ) + αzf ′′ ( z ) ≺ h ( z ) , z ∈ D } , where h is a convex function have been investigated by several authors [6, 23, 34, 36, 38]. On thebasis of the above discussed works, we consider a unified class of all functions f ∈ A suchthat(1.3) f ′ ( z ) + αzf ′′ ( z ) ≺ φ ( z )for z ∈ D and where α ∈ C with Re α ≥
0. The class of such functions is denoted by R ( α, φ ). Since f ∈ R ( α, φ ), f ′ ( z ) + αzf ′′ ( z ) = φ ( e iθ ), θ ∈ [0 , π ), it is observed that f ′ ( z ) + αzf ′′ ( z ) = [(1 − α ) f ( z ) + α ( zf ′ ( z ))] ′ . Also, we have zf ′ ( z ) = f ( z ) ∗ z (1 − z ) and f ( z ) = f ( z ) ∗ z − z . Thus, f ′ ( z ) + αzf ′′ ( z ) = (cid:18) (1 − α ) f ( z ) ∗ z − z + αf ( z ) ∗ z (1 − z ) (cid:19) ′ = (cid:18) f ( z ) ∗ (cid:18) (1 − α ) z − z + α z (1 − z ) (cid:19) (cid:19) ′ . Therefore, we conclude that (cid:18) f ( z ) ∗ z − z (1 − α )(1 − z ) (cid:19) ′ − φ ( e iθ ) = 0or equivalently 1 z (cid:18) f ( z ) ∗ z + z (2 α − − z ) (cid:19) = φ ( e iθ )which is the necessary and sufficient conditions for a function f ∈ A to be in the class R ( α, φ ).In this paper, we compute the distortion, growth inequalities for a function f in theclass R ( α, φ ). The sharp bounds on initial logarithmic coefficients for such functions arealso obtained. Next, we obtain the bounds on initial inverse coefficients of the function f ∈ R ( α, φ ) as well as bounds on Fekete Szeg¨o functional and second Hankel determinant.2. Distortion and Growth Theorem
The first theorem proves the distortion theorem of the functions f belonging to R ( α, φ ). Theorem 2.1.
Let α ∈ C , Re α ≥ and the function φ be defined by (1.2) . If the function f ∈ R ( α, φ ) , then ∞ X n =1 | B n | ( − r ) n n Re α + 1 ≤ | f ′ ( z ) | ≤ ∞ X n =1 | B n | r n n Re α + 1 for | z | < r < . The result is sharp. STIMATES OF NORMALIZED ANALYTIC FUNCTIONS 3
We make use of the following lemma in order to prove some of our results.
Lemma 2.2. [7, Lemma 2, p. 192]
Let h be a convex function with Re γ ≥ . If p ( z ) isregular in D and p (0) = h (0) , then (2.1) p ( z ) + zp ′ ( z ) γ ≺ h ( z ) implies that p ( z ) ≺ q ( z ) ≺ h ( z ) , where q ( z ) = γz − γ Z z h ( t ) t γ − dt. The function q is convex and the best dominant.Proof of Theorem 2.1. Let the fuction f be in the class R ( α, φ ). Then f ′ ( z ) + αzf ′′ ( z ) ≺ φ ( z ) . For p ( z ) = f ′ ( z ) and γ = 1 /α , Lemma 2.2 yields f ′ ( z ) ≺ α z − /α Z z φ ( t ) t /α − dt. On taking t = zζ α in (2), we get(2.2) f ′ ( z ) ≺ Z φ ( zζ α ) dζ . Since the function φ is symmetric with respect to real axis, φ ( z ) has real coefficients. Also φ ′ (0) > φ ′ ( x ) is increasing on (0 , | z = r | Re φ ( z ) = φ ( − r ) and max | z = r | Re φ ( z ) = φ ( r ) . Using (2.3) and (2.2) for | z | = r , we have | f ′ ( z ) | ≥ Re f ′ ( z ) ≥ min | z | = r Re f ′ ( z ) ≥ min | z | = r Re Z φ ( zζ α ) dζ = Z min | z | = r Re φ ( zζ α ) dζ (2.4) = Z φ ( − rζ Re α ) dζ . Similarly, we have(2.5) | f ′ ( z ) | ≤ Z φ ( rζ Re α ) dζ . Since φ ( zζ Re α ) = 1 + B zζ Re α + B z ζ α + · · · , a simple calculation yields Z φ ( zζ Re α ) dζ = Z (1 + B zζ Re α + B z ζ α + B z ζ α + · · · ) dζ = 1 + B z Re α + 1 + B z α + 1 + B z α + 1 + · · · = 1 + ∞ X n =1 B n z n n Re α + 1 . (2.6) S. ANAND, N.K. JAIN, AND S. KUMAR
From (2.4), (2.5) and (2.6) the result follows. The result is sharp for the function f : D → C defined by(2.7) f ( z ) = z + ∞ X n =1 B n z n +1 ( n + 1)(1 + n Re α ) . Theorem 2.3.
Let α ∈ C such that Re α ≥ and the function φ be as in (1.2). Then forthe function f ∈ R ( α, φ ) , we have ∞ X n =1 | B n | ( − r ) n ( n + 1)( n Re α + 1) ≤ | f ( z ) || z | ≤ ∞ X n =1 | B n | r n ( n + 1)( n Re α + 1) ( | z | < r < . Proof.
Let(2.8) H ( z ) = Z φ ( zζ α ) dζ and Φ α ( z ) = Z − zt α dt = ∞ X n =0 z n nα . From [32, Theorem 5, p.113], it is noted that Φ α ( z ) is convex with Re α ≥
0. Also,Φ α ( z ) ∗ φ ( z ) = ∞ X n =0 z n nα ! ∗ ∞ X n =1 B n z n ! = 1 + ∞ X n =1 (cid:18) B n nα (cid:19) z n . It view of above and (2.6), we haveΦ α ( z ) ∗ φ ( z ) = Z φ ( zζ α ) dζ = H ( z ) . Since convolution of two convex functions is convex, the function H is convex and H (0) = 1.Putting γ = 1 and h ( z ) = H ( z ) in Lemma 2.2, we get(2.9) p ( z ) ≺ z Z z H ( t ) dt Using (2.8), substituting t = zσ and p ( z ) = f ( z ) /z in (2.9), we have(2.10) f ( z ) z ≺ Z Z φ ( zσζ α ) dσdζ . Let h ( z ) = f ( z ) /z . Then (2.3) together with (2.10) yields(2.11) | h ( z ) | ≥ min | z | = r Re h ( z ) ≥ Z Z min | z | = r φ ( zσζ α ) dσdζ = Z Z φ ( − rσζ Re α ) dσdζ . and(2.12) | h ( z ) | ≤ max | z | = r Re h ( z ) ≤ Z Z max | z | = r φ ( zσζ α ) dσdζ = Z Z φ ( rσζ Re α ) dσdζ . STIMATES OF NORMALIZED ANALYTIC FUNCTIONS 5
A simple calculation shows that Z Z φ ( zσζ Re α ) dσdζ = Z ∞ X n =1 (cid:18) B n n Re α (cid:19) ( zσ ) n ! dσ = 1 + ∞ X n =1 B n z n ( n + 1)(1 + n Re α ) . (2.13)Now, the result follows from (2.11), (2.12) and (2.13). The result is sharp for the function f given by (2.7). Remark . Letting φ ( z ) = (1 − (1 − β ) z ) / (1 − z ), where β < α = 1, Theorem2.3 reduces to a result due to Silverman [33, Corollary 2, p.250]. Further, for φ ( z ) =(1 − (1 − β ) z ) / (1 − z ), where β < α >
0, Theorem 2.5 yields [6, Corollary 3, p.178].
Theorem 2.5.
Let α ∈ C such that Re α ≥ and the function φ be as in (1.2) such that f ∈ R ( α, φ ) . Then | a n | ≤ B | n + n ( n − α | , for all n ≥ . (2.14) Proof.
For p ( z ) = 1 + P ∞ n =1 p n z n ∈ P , set f ′ ( z ) + αzf ′′ ( z ) = p ( z ), z ∈ D . Since f ∈ R ( α, φ ), p ( z ) ≺ φ ( z ). A simple calculation gives(2.15) f ′ ( z ) + αzf ′′ ( z ) = 1 + ∞ X n =2 [ n + n ( n − α ] a n z n − = 1 + ∞ X n =1 p n z n . On comparing the coefficients of z n − , we get( n + n ( n − α ) a n = p n − , for all n ≥ . By making use of [31, Theorem X, p.70], we get | p n | ≤ B , for all n ≥ . Hence, weget the desired inequality. The inequality (2.14) is sharp for the function f n given by f ′ n ( z ) + αzf ′′ n ( z ) = φ ( z n − ) . Remark . On taking φ ( z ) = (1 − (1 − β ) z ) / (1 − z ), where β < α = 1, Theorem2.5 yields a result due to Silverman [33, Corollary 1, p.250]. Further, letting φ ( z ) = (1 − (1 − β ) z ) / (1 − z ), where β < α > Bounds on Initial Logarithmic Coefficient
For a function f ∈ S , the logarithmic coefficients γ n are defined by the following seriesexpansion:(3.1) log f ( z ) z = 2 ∞ X n =1 γ n z n , z ∈ D \ { } , log 1 := 0 . On comparing the coefficients of z on both the sides, we get the initial logarithmic coeffi-cients γ = 12 a , γ = 12 (cid:18) a − a (cid:19) γ = 12 (cid:18) a − a a + 13 a (cid:19) (3.2) S. ANAND, N.K. JAIN, AND S. KUMAR
In 1979, the authors [5] showed that the logarithmic coefficients γ n of every function f ∈ S satisfy the inequality ∞ P n =1 | γ n | ≤ π /
6, where the equality holds if and only if the function f is rotation of the Koebe function k ( z ) = z (1 − e iθ ) − for each θ . The n th logarithmiccoefficient of k ( z ) is γ n = e inθ /n for each θ and n ≥
1. In [39], the logarithmic coefficients γ n of each close-to-convex function f ∈ S is bounded by ( A log n ) /n where A is an absoluteconstant. In 2018, the authors [4, 27] obtained the bounds on logarithmic coefficients ofcertain subclasses of the class of close-to-convex functions. Recently, Adegani et. al. [1]investigated the bounds for the initial logarithmic coefficients of the generalized classes S ∗ ( φ ) and C ( φ ). To find the bounds on initial logarithmic coefficient for class R ( α, φ ), weshall use the following two lemmas. Lemma 3.1. [25, p.172]
Assume that w is a Schwarz function so that w ( z ) = ∞ P n =1 c n z n .Then | c | ≤ and | c n | ≤ − | c | , n = 2 , , · · · . Lemma 3.2. [28, Theorem 1]
Let w ( z ) = ∞ P n =1 c n z n be the Schwarz function. Then for anyreal numbers q and q , the following sharp inequality holds: | c + q c c + q c | ≤ H ( q ; q ) , where H ( q ; q ) = , if ( q , q ) ∈ D ∪ D ∪ { (2 , } , | q | , if ( q , q ) ∈ ∪ k =1 D k ,
23 ( | q | + 1) (cid:18) | q | | q | + 1 + q ) (cid:19) , if ( q , q ) ∈ D ∪ D ,q (cid:18) q − q − q (cid:19) (cid:18) q − q − (cid:19) , if ( q , q ) ∈ D ∪ D \ { (2 , } ,
23 ( | q | − (cid:18) | q | − | q | − − q ) (cid:19) , if ( q , q ) ∈ D STIMATES OF NORMALIZED ANALYTIC FUNCTIONS 7 and for k = 1 , , · · · , the sets D k are defined as follows: D = (cid:26) ( q , q ) : | q | ≤ , | q | ≤ (cid:27) ,D = (cid:26) ( q , q ) : 12 ≤ | q | ≤ ,
427 ( | q | + 1) − ( | q | + 1) ≤ q ≤ (cid:27) ,D = (cid:26) ( q , q ) : | q | ≤ , q ≤ − (cid:27) ,D = (cid:26) ( q , q ) : | q | ≥ , q ≤ −
23 ( | q | + 1) (cid:27) ,D = { ( q , q ) : | q | ≤ , q ≥ } ,D = (cid:26) ( q , q ) : 2 ≤ | q | ≤ , q ≥
112 ( q + 8) (cid:27) ,D = (cid:26) ( q , q ) : | q | ≥ , q ≥
23 ( | q | − (cid:27) ,D = (cid:26) ( q , q ) : 12 ≤ | q | ≤ , −
23 ( | q | + 1) ≤ q ≤
427 ( | q | + 1) − ( | q | + 1) (cid:27) ,D = (cid:26) ( q , q ) : | q | ≥ , −
23 ( | q | + 1) ≤ q ≤ | q | ( | q | + 1) q + 2 | q | + 4 (cid:27) ,D = (cid:26) ( q , q ) : 2 ≤ | q | ≤ , | q | ( | q | + 1) q + 2 | q | + 4 ≤ q ≤
112 ( q + 8) (cid:27) ,D = (cid:26) ( q , q ) : | q | ≥ , | q | ( | q | + 1) q + 2 | q | + 4 ≤ q ≤ | q | ( | q | − q − | q | + 4 (cid:27) ,D = (cid:26) ( q , q ) : | q | ≥ , | q | ( | q | − q − | q | + 4 ≤ q ≤
23 ( | q | − (cid:27) . Theorem 3.3.
Let α ∈ C such that Re α ≥ and the function φ be as in (1.2). Suppose f ∈ R ( α, φ ) , then the initial logarithmic coefficients of f satisfy the following inequalities: (i) | γ | ≤ B | α | . (ii) | γ | ≤ B | α | , if | α ) B − α ) B | ≤ B | (1 + α ) || α ) B − α ) B | | (1 + α ) || α | , if | α ) B − α ) B | > B | (1 + α ) | (iii) If B , B , B and α are real numbers, then | γ | ≤ B α ) H ( q ; q ) , where H ( q ; q ) is given in Lemma 3.2 such that q = 2 B B − B (1 + 3 α )3(1 + α )(1 + 2 α ) S. ANAND, N.K. JAIN, AND S. KUMAR and q = B B − B (1 + 3 α )3(1 + α )(1 + 2 α ) + B (1 + 3 α )6(1 + α ) . The bounds for γ and γ are sharp.Proof. Let f ( z ) = z + ∞ P n =2 a n z n ∈ R ( α, φ ) where φ is given by (1.2). Then f ′ ( z ) + αzf ′′ ( z ) = φ ( w ( z )) for z ∈ D , where w ( z ) = ∞ P n =1 c n z n is the Schwarz function. A simple calculationyields f ′ ( z ) + αzf ′′ ( z ) = 1 + B c z + ( B c + B c ) z + ( B c + 2 c c B + B c ) z + · · · . On comparing the coefficients of z , we obtain2(1 + α ) a = B c , α ) a = B c + B c and4(1 + 3 α ) a = B c + 2 B c c + B c . On substituting the above values of a i ( i = 1 , ,
3) in (3.2), we get γ = B c α ) ,γ = 8(1 + α ) B c + (8(1 + α ) B − α ) B ) c α ) (1 + 2 α ) ,γ = B α ) c + (cid:18) B α ) − B α )(1 + 2 α ) (cid:19) c c + (cid:18) B α ) − B B α )(1 + 2 α ) + B α ) (cid:19) c . (3.3)By using Lemma 3.1, we get the desired best possible estimate on γ . The bound is sharpfor | c | = 1. | γ | = (cid:12)(cid:12)(cid:12)(cid:12) B α ) c + 8(1 + α ) B − α ) B α ) (1 + 2 α ) c (cid:12)(cid:12)(cid:12)(cid:12) ≤ B | α | | c | + | α ) B − α ) B | | (1 + α ) || α | | c |≤ B | α | (1 − | c | ) + | α ) B − α ) B ) | | (1 + α ) || α | | c | = B | α | + (cid:18) | α ) B − α ) B | | (1 + α ) || α | − B | α | (cid:19) | c |≤ B | α | , if | α ) B − α ) B | | (1 + α ) || α | ≤ B | α || α ) B − α ) B | | (1 + α ) || α | , if | α ) B − α ) B | | (1 + α ) || α | > B | α | These bounds are sharp for | c | = 0 and | c | = 1, respectively. STIMATES OF NORMALIZED ANALYTIC FUNCTIONS 9
The third inequality follows by Lemma 3.1. Using Lemma 3.2 for γ , we obtain | γ | = (cid:12)(cid:12)(cid:12)(cid:12) B α ) c + (cid:18) B α ) − B α )(1 + 2 α ) (cid:19) c c + (cid:18) B α ) − B B α )(1 + 2 α ) + B α ) (cid:19) c (cid:12)(cid:12)(cid:12)(cid:12) = B α ) | c + c c q + c q |≤ B α ) H ( q ; q ) , where q = 2 (cid:18) B B − B (1 + 3 α )3(1 + α )(1 + 2 α ) (cid:19) and q = B B − B (1 + 3 α )3(1 + α )(1 + 2 α ) + B (1 + 3 α )6(1 + α ) .This completes the proof.On taking φ ( z ) = e z , φ ( z ) = √ z and φ ( z ) = (1 + z ) / (1 − z ), respectively in Theorem3.3 the following corollaries follows immediately. Corollary 3.4.
Let α ∈ C such that Re α ≥ and φ ( z ) = e z . Suppose f ∈ R ( α, φ ) , thenthe initial logarithmic coefficients of f satisfy the following inequalities: (i) | γ | ≤ | α | (ii) | γ | ≤ | α | (iii) | γ | ≤ α ) H ( q ; q ) ,where H ( q ; q ) is given in Lemma 3.2 such that q = 1 + 3 α (1 + 2 α )3(1 + α )(1 + 2 α ) and q = 16 − (1 + 3 α )3(1 + α )(1 + 2 α ) + (1 + 3 α )6(1 + α ) . Corollary 3.5.
Suppose that f ∈ R ( α, φ ) where Re α ≥ and φ ( z ) = √ z , then theinitial logarithmic coefficients of f satisfy the following inequalities: (i) | γ | ≤ | α | (ii) | γ | ≤ | α | (iii) | γ | ≤ α ) H ( q ; q ) ,where H ( q ; q ) is given in Lemma 3.2 such that q = − − (1 + 3 α )3(1 + α )(1 + 2 α ) and q = 18 + 1 + 3 α α )(1 + 2 α ) + 1 + 3 α α ) . Corollary 3.6.
Let the function f ∈ R ( α, φ ) where Re α ≥ and φ ( z ) = (1 + z ) / (1 − z ) ,then the initial logarithmic coefficients of f satisfy the following inequalities: (i) | γ | ≤ | α | (ii) | γ | ≤ | α | (iii) | γ | ≤ α ) H ( q ; q ) ,where H ( q ; q ) is given in Lemma 3.2 such that q = 2(1 + 3 α (1 + 2 α ))3(1 + α )(1 + 2 α ) and q = 1 − α )3(1 + α )(1 + 2 α ) + 2(1 + 3 α )3(1 + α ) . Inverse Coefficient Estimates
From the Koebe one quarter theorem, the image of D under a function f ∈ S containsa disk of radius 1 /
4. Thus for every univalent function f there exist inverse function f − such that f − ( f ( z )) = z for z ∈ D and f ( f − ( ω )) = ω for | ω | < r ( f ) where r ( f ) ≥ / f − has the Taylor series expansion f − ( ω ) = ω + A ω + A ω + · · · insome neighborhood of origin. In 1923, L¨owner [20] initiated the problem of estimating thecoefficients of inverse function and investigated the coefficient estimates for inverse function f ∈ S . Later on, this lead to the study of inverse coefficient problem for several subclassesof S by various authors [2, 17, 18, 19, 28]. In [12, 15], authors obtained the initial inversecoefficients for the well known classes C and S ∗ ( α ) (0 ≤ α ≤ Lemma 4.1. [17, Lemma 3, p.254] If p ( z ) = 1 + p z + p z + . . . is a function in the class P , then for any complex number ν , | p − νp | ≤ { , | ν − |} . Theorem 4.2.
Let α ∈ C such that Re α ≥ and the function φ defined by (1.2). Iffunction f ( z ) = z + ∞ P n =2 a n z n ∈ R ( α, φ ) and f − ( ω ) = ω + ∞ P n =2 A n ω n for all ω in someneighbourhood of the origin, then (i) | A | ≤ B | α | , (ii) | A | ≤ B | α | max { , | µ |} , where µ = 3 B (1 + 2 α )2(1 + α ) − B B (iii) If B , B , B and α are real numbers, then | A | ≤ B α ) H ( q ; q ) , STIMATES OF NORMALIZED ANALYTIC FUNCTIONS 11 where H ( q ; q ) is given in Lemma 3.2 such that q = 2 (cid:18) B B − B (1 + 3 α )3(1 + α )(1 + 2 α ) (cid:19) and q = B B − B (1 + 3 α )3(1 + α )(1 + 2 α ) + 5 B (1 + 3 α )2(1 + α ) . Proof.
Let f ∈ R ( α, φ ). Then(4.1) f ′ ( z ) + αzf ′′ ( z ) = φ ( w ( z )) , where w ( z ) is the analytic function w with w (0) = 0 and | w ( z ) | <
1. Let p ( z ) = 1 + w ( z )1 − w ( z ) = 1 + p z + p z + p z + · · · . Since w : D → D is analytic thus p is a function with positive real part and(4.2) w ( z ) = p ( z ) − p ( z ) + 1 = 12 p z + 12 ( p − p z + 18 ( p − p p + 4 p ) z + · · · . Then φ ( w ( z )) = 1 + B p z + (cid:18) B p + 12 B ( p − p ) (cid:19) z + 18 (( B − B + B ) p + 4( − B + B ) p p + 4 B p ) z + · · · . (4.3)Using expressions (4.3) and (4.1), we obtain2(1 + α ) a = B p , α ) a = 14 B p + 12 B ( p − p ) and4(1 + 3 α ) a = 18 (( B − B + B ) p + 4( − B + B ) p p + 4 B p ) . (4.4)Since f − ( ω ) = ω + A ω + A ω + A ω + · · · in some neighbourhood of origin, so we have f ( f − ( ω )) = ω . That is ω = f − ( ω ) + a ( f − ( ω )) + a ( f − ( ω )) + · · · = ω + A ω + A ω + A ω + · · · + a ( ω + A ω + A ω + A ω + · · · ) + a ( ω + A ω + A ω + A ω + · · · ) . A simple calculation gives the following realtions: A = − a ,A = 2 a − a and A = − a + 5 a a − a . (4.5) On subsituting the values of a i from (4.4) into (4.5) and a simple calculation yields A = − B α ) p ,A = − B α ) p + (cid:18) B α ) − B α ) + B α ) (cid:19) p In (4.2), on taking c = p c = 12 ( p − p c = 18 ( p − p p + 4 p ) and so on we get,2(1 + α ) a = B c , α ) a = B c + B c and4(1 + 3 α ) a = B c + 2 B c c + B c . (4.6)On substituting the values of a i from (4.6) in (4.5), we obtain A = − B α ) c + (cid:18) B α )(1 + 2 α ) − B α ) (cid:19) c c + (cid:18) − B α ) + 5 B B α )(1 + 2 α ) + − B α ) (cid:19) c . Since | p | ≤
2, we have | A | ≤ B | α | . Consider | A | = B | α | (cid:12)(cid:12)(cid:12)(cid:12) p − (cid:18) B (1 + 2 α )4(1 + α ) − B B + 12 (cid:19) p (cid:12)(cid:12)(cid:12)(cid:12) . Then by Lemma 4.1, we get the desired estimate. Using Lemma 3.2 for A , we obtain | A | = (cid:12)(cid:12)(cid:12)(cid:12) − B α ) c + (cid:18) − B α ) + 5 B α )(1 + 2 α ) (cid:19) c c + (cid:18) − B α ) + 5 B B α )(1 + 2 α ) − B α ) (cid:19) c (cid:12)(cid:12)(cid:12)(cid:12) = B α ) | c + c c q + c q |≤ B α ) H ( q ; q ) , where q = 2 (cid:18) B B − B (1 + 3 α )3(1 + α )(1 + 2 α ) (cid:19) and q = B B − B (1 + 3 α )3(1 + α )(1 + 2 α ) + 5 B (1 + 3 α )2(1 + α ) .The following corollaries are an immediate consequence of the Theorem 4.2 for φ ( z ) = e z , φ ( z ) = √ z and φ ( z ) = (1 + z ) / (1 − z ), respectively. Corollary 4.3.
Let α ∈ C such that Re α ≥ and φ ( z ) = e z . If the function f ( z ) = z + ∞ P n =2 a n z n ∈ R ( α, φ ) and f − ( ω ) = ω + ∞ P n =2 A n ω n for all ω in some neighbourhood of theorigin, then STIMATES OF NORMALIZED ANALYTIC FUNCTIONS 13 (i) | A | ≤ | α | , (ii) | A | ≤ | α | max (cid:26) , (cid:12)(cid:12)(cid:12)(cid:12) α )2(1 + α ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) (iii) | A | ≤ α ) H ( q ; q ) ,where α is real and H ( q ; q ) is given in Lemma 3.2 such that q = − α ( − α )3(1 + α )(1 + 2 α ) and q = 16 − α )3(1 + α )(1 + 2 α ) + 5(1 + 3 α )2(1 + α ) . Corollary 4.4.
Suppose that the function f ( z ) = z + ∞ P n =2 a n z n ∈ R ( α, φ ) , where Re α ≥ and φ ( z ) = √ z and f − ( ω ) = ω + ∞ P n =2 A n ω n for all ω in some neighbourhood of theorigin, then (i) | A | ≤ | α | , (ii) | A | ≤ | α | max (cid:26) , (cid:12)(cid:12)(cid:12)(cid:12) α )4(1 + α ) + 14 (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) , (iii) | A | ≤ α ) H ( q ; q ) ,where α is real and H ( q ; q ) is given in Lemma 3.2 such that q = − − α )3(1 + α )(1 + 2 α ) and q = 18 + 5(1 + 3 α )12(1 + α )(1 + 2 α ) + 5(1 + 3 α )8(1 + α ) . Hankel Determinant
The problem involving coefficient bounds have attracted the interest of many authorsin particular to second Hankel determinants and Fekete Szeg¨o functional. The coefficientfunctional | a − µa | where µ is a complex number is called the Fekete Szeg¨o functional. TheFekete Szeg¨o problem involves maximizing the functional | a − µa | . Some authors haveinvestigated the Fekete Szeg¨o problem for the coefficients of inverse function [2, 24, 37]. TheHankel determinant | H (1) | = | a − a | is a particular case of the Fekete Szeg¨o functionaland H (2) = | a a − a | is called the second Hankel determinants. In 2013, Lee et. al. [16] obtained the bounds for the second Hankel determinant for the unified class of Ma-Minda starlike and convex functions. The authors [37] estimated the bounds on secondHankel determinant for the class of strongly convex functions of order α using the inversecoefficients. One may refer to [14, 8, 13, 9, 26] for more details. In the present section, weshall determine the Fekete Szeg¨o functional for the inverse coefficient. Theorem 5.1.
Suppose α ∈ C such that Re α ≥ and the function φ defined by (1.2).Let f ∈ R ( α, φ ) and f − ( ω ) = ω + ∞ P n =2 A n ω n for all ω in some neighbourhood of the origin.Then for any complex number µ , we have | A − µA | ≤ B | α | max (cid:26) , (cid:12)(cid:12)(cid:12)(cid:12) B (1 + 2 α )4(1 + α ) ( µ −
2) + B B (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) . Proof.
In view of euqations (4.4) and (4.5), we get | A − µA | = (cid:12)(cid:12)(cid:12)(cid:12) − B α ) p + (cid:18)(cid:18) B α ) + B − B α ) (cid:19) − µ B α ) (cid:19) p (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) B α ) (cid:18) p + (cid:18) − B (1 + 2 α )4(1 + α ) + B B −
12 + µ B (1 + 2 α )8(1 + α ) (cid:19) p (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) B α ) ( p − νp ) (cid:12)(cid:12)(cid:12)(cid:12) , where ν = 3 B (1 + 2 α )8(1 + α ) (2 − µ ) − B B + 12 . By Lemma 4.1, we get the required result. Theorem 5.2.
Let α ∈ C such that Re α ≥ and the function φ defined by (1.2). Supposefunction f in R ( α, φ ) and f − ( ω ) = ω + ∞ P n =2 A n ω n for all ω in some neighbourhood of theorigin. (i) If B , B and B satisfy the conditions d ≤ d , d ≤ B | α | | (1 + 2 α ) | , then | A A − A | ≤ B | (1 + 2 α ) | . (ii) If B , B and B satisfy the conditions d ≥ d , d − d − B | α | ≥ or d ≤ d , d ≥ B | α | | (1 + 2 α ) | , then | A A − A | ≤ B d | α | . (iii) If B , B and B satisfy the conditions d > d , d − d − B | α | ≤ , STIMATES OF NORMALIZED ANALYTIC FUNCTIONS 15 then | A A − A | ≤ B | α | (cid:18) B | α | | (1 + 2 α ) | d − B | α | d − d − B | (1 + 3 α ) | (cid:19) d − d − B | α | + B | α | | (1 + 2 α ) | , where d = B | (1 + α ) | + B | α | B | (1 + 2 α ) | + B | B | | α || α | + | B | | α | d = B | α || α | + | B | | α | + 2 | B || α | | (1 + 2 α ) | and d = 8 B | α | | (1 + 2 α ) | − B | α | . In the proof of this result, the following lemma is needed.
Lemma 5.3. [17, Lemma 2, p.254] If p ( z ) = 1 + ∞ P n =1 p n z n ∈ P , then p = p + x (4 − p )4 p = p + 2 p (4 − p ) x − p (4 − p ) x + 2(4 − p )(1 − | x | ) z for some x and z such that | x | ≤ and | z | ≤ .Remark . For real numbers P , Q and R , a standard computation gives(5.1) max ≤ t ≤ ( P t + Qt + R ) = R, Q ≤ , P ≤ − Q/ P + 4 Q + R, Q ≥ , P ≥ − Q/ Q ≤ , P ≥ − Q/ P R − Q P , Q > , P ≤ − Q/ Proof of Theorem 5.2 .
It follows from equations (4.4) and (4.5) that A A − A = − B p α ) (cid:18) − B α ) p + (cid:18) B α )(1 + 2 α ) + B − B α ) (cid:19) p p + (cid:18) − B α ) − B ( B − B )48(1 + α )(1 + 2 α ) − B − B + B α ) (cid:19) p (cid:19) − (cid:18)(cid:18) B α ) + ( B − B )12(1 + 2 α ) (cid:19) p − B p α ) + (cid:19) = p (cid:18) B α ) + B ( B − B )192(1 + α ) (1 + 2 α ) + B ( B − B + B )128(1 + α )(1 + 3 α ) − ( B − B ) α ) (cid:19) + B p p α )(1 + 3 α ) − B p α ) + p p (cid:18) − B α ) (1 + 2 α ) − B ( B − B )32(1 + α )(1 + 3 α ) + B ( B − B )36(1 + 2 α ) (cid:19) . In view of Lemma 5.3, we obtain A A − A = B α ) (cid:20) p (cid:18) B α ) − B (1 + α )9 B (1 + 2 α ) − B B α )(1 + 2 α ) + B α ) (cid:19) + 2 p (4 − p ) x (cid:18) − B α )(1 + 2 α ) + B α ) − B (1 + α )9(1 + 2 α ) (cid:19) +(4 − p ) x p (cid:18) − B α ) (cid:19) − (4 − p ) x (cid:18) B (1 + α )9(1 + 2 α ) (cid:19) +2 p (4 − p )(1 − | x | ) z B α ) (cid:21) . Since | p | ≤ p > p ∈ [0 , p = p and | x | = γ in the above expression, we get | A A − A | ≤ B | α | (cid:20) p (cid:18) B | (1 + α ) | + B | α | B | (1 + 2 α ) | + B | B | | α || α | + | B | | α | (cid:19) + 2 p (4 − p ) γ (cid:18) B | α || α | + | B | | α | + | B || α | | (1 + 2 α ) | (cid:19) +(4 − p ) γ p (cid:18) B | α | (cid:19) + (4 − p ) γ (cid:18) B | α | | (1 + 2 α ) | (cid:19) +2 p (4 − p )(1 − γ ) | z | B | α | (cid:21) . =: G ( p, γ ) . Let p be fixed. Using the first derivative test in the region Ω = { ( p, γ ) : 0 ≤ p ≤ , ≤ γ ≤ } we get that G ( p, γ ) is an increasing function of γ where γ ∈ [0 , p ∈ [0 , ≤ γ ≤ G ( p, γ ) = G ( p,
1) =: F ( p ), where F ( p ) = B | α | (cid:20) p (cid:18) B | (1 + α ) | + B | α | B | (1 + 2 α ) | + B | B | | α || α | + | B | | α |− B | α || α | − | B | | α | − | B || α | | (1 + 2 α ) | − B | α | + B | α | | (1 + 2 α ) | (cid:19) + p (cid:18) B | α || α | + | B || α | + 8 | B || α | | (1 + 2 α ) | + B | α | − B | α | | (1 + 2 α ) | (cid:19) + 16 B | α | | (1 + 2 α ) | (cid:21) . STIMATES OF NORMALIZED ANALYTIC FUNCTIONS 17
Let P = B | α | (cid:20)(cid:18) B | (1 + α ) | + B | α | B | (1 + 2 α ) | + B | B | | α || α | + | B | | α | (cid:19) − (cid:18) B | α || α | + | B | | α | + 2 | B || α | | (1 + 2 α ) | (cid:19) + (cid:18) − B | α | + B | α | | (1 + 2 α ) | (cid:19)(cid:21) ,Q = B | α | (cid:20) (cid:18) B | α || α | + B | α | + 2 | B || α | | (1 + 2 α ) | (cid:19) − (cid:18) B | α | | (1 + 2 α ) | − B | α | (cid:19)(cid:21) ,R = B | (1 + 2 α ) | and t = p . Then F ( t ) = P t + Qt + R . Using the inequality (5.1), we get the required result. References [1] E. A. Adegani, N. E. Cho, and M. Jafari, Logarithmic Coefficients for Univalent Functions Defined bySubordination. Mathematics 2019, 7, 408.[2] R. M. Ali, Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc. (2) (2003), no. 1, 63–71.[3] R. M. Ali, S. K. Lee, K. G. Subramanian, and A. Swaminathan, A third-order differential equationand starlikeness of a double integral operator, Abstr. Appl. Anal. , Art. ID 901235, 10 pp.[4] M. F. Ali and A. Vasudevarao, On logarithmic coefficients of some close-to-convex functions, Proc.Amer. Math. Soc. (2018), no. 3, 1131–1142.[5] P. L. Duren and Y. J. Leung, Logarithmic coefficients of univalent functions, J. Analyse Math. (1979), 36–43 (1980).[6] C. Y. Gao and S. Q. Zhou, Certain subclass of starlike functions, Appl. Math. Comput. (2007),no. 1, 176–182.[7] D. J. Hallenbeck and S. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math. Soc. (1975), 191–195.[8] T. Hayami and S. Owa, Generalized Hankel determinant for certain classes, Int. J. Math. Anal. (Ruse) (2010), no. 49-52, 2573–2585.[9] W. K. Hayman, On the second Hankel determinant of mean univalent functions, Proc. London Math.Soc. (3) (1968), 77–94.[10] N. K. Jain and S. Yadav, Bohr Radius for Certain Analytic Functions, In: Deo N., Gupta V., AcuA., Agrawal P. (eds) Mathematical Analysis I: Approximation Theory. ICRAPAM 2018. SpringerProceedings in Mathematics & Statistics, vol 306. Springer, Singapore, 2020.[11] W. Janowski, Extremal problems for a family of functions with positive real part and for some relatedfamilies, Ann. Polon. Math. (1970/1971), 159–177.[12] G. P. Kapoor and A. K. Mishra, Coefficient estimates for inverses of starlike functions of positive order,J. Math. Anal. Appl. (2007), no. 2, 922–934.[13] V. Kumar, S. Kumar and V. Ravichandran (2020) Third Hankel Determinant for Certain Classesof Analytic Functions. In: Deo N., Gupta V., Acu A., Agrawal P. (eds) Mathematical Analysis I:Approximation Theory. ICRAPAM 2018. Springer Proceedings in Mathematics and Statistics, vol 306,pp 223-231. Springer, Singapore.[14] S. Kumar, V. Ravichandran and S. Verma, Initial coefficients of starlike functions with real coefficients,Bull. Iranian Math. Soc. (2017), no. 6, 1837–1854.[15] J. G. Krzy˙z, R. J. Libera and E. Z lotkiewicz, Coefficients of inverses of regular starlike functions, Ann.Univ. Mariae Curie-Sk lodowska Sect. A (1979), 103–110 (1981).[16] S. K. Lee, V. Ravichandran and S. Supramaniam, Bounds for the second Hankel determinant of certainunivalent functions, J. Inequal. Appl. , 2013:281, 17 pp. [17] R. J. Libera and E. J. Z lotkiewicz, Coefficient bounds for the inverse of a function with derivative in P , Proc. Amer. Math. Soc. (1983), no. 2, 251–257.[18] R. J. Libera and E. J. Z lotkiewicz, The coefficients of the inverse of an odd convex function, RockyMountain J. Math. (1985), no. 3, 677–683.[19] R. J. Libera and E. J. Z lotkiewicz, L¨owner’s inverse coefficients theorem for starlike functions, Amer.Math. Monthly (1992), no. 1, 49–50.[20] K. Lowner, Untersuchungen uber schlichte konforme Abbildungen des Einheitskreises. I, Math. Ann.89 (1923), no. 1-2, 103121.[21] W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, in Pro-ceedings of the Conference on Complex Analysis (Tianjin, 1992) , 157–169, Conf. Proc. Lecture NotesAnal., I, Int. Press, Cambridge, MA.[22] R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associatedwith exponential function, Bull. Malays. Math. Sci. Soc. (2015), no. 1, 365–386.[23] M. A. Nasr and M. K. Aouf, Bounded starlike functions of complex order, Proc. Indian Acad. Sci.Math. Sci. (1983), no. 2, 97–102.[24] A. Naz, S. Kumar and V. Ravichandran, Coefficients of the inverse functions and radius estimates ofcertain starlike functions, preprint.[25] Z. Nehari, Conformal mapping , McGraw-Hill Book Co., Inc., New York, Toronto, London, 1952.[26] J. W. Noonan and D. K. Thomas, On the second Hankel determinant of areally mean p -valent functions,Trans. Amer. Math. Soc. (1976), 337–346.[27] U. Pranav Kumar and A. Vasudevarao, Logarithmic coefficients for certain subclasses of close-to-convexfunctions, Monatsh. Math. (2018), no. 3, 543–563.[28] D. V. Prokhorov and J. Szynal, Inverse coefficients for ( α, β )-convex functions, Ann. Univ. MariaeCurie-Sk lodowska Sect. A (1981), 125–143 (1984).[29] V. Ravichandran and S. Verma, Estimates for coefficients of certain analytic functions, Filomat (2017), no. 11, 3539–3552.[30] M. S. Robertson, Certain classes of starlike functions, Michigan Math. J. (1985), no. 2, 135–140.[31] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. (2) (1943),48–82.[32] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. (1975), 109–115.[33] H. Silverman, A class of bounded starlike functions, Internat. J. Math. Math. Sci. (1994), no. 2,249–252.[34] R. Singh and S. Singh, Convolution properties of a class of starlike functions, Proc. Amer. Math. Soc. (1989), no. 1, 145–152.[35] J. Sok´o l and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions,Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 19 (1996), 101–105.[36] H. M. Srivastava, D. R˘aducanu and P. Zaprawa, A certain subclass of analytic functions defined bymeans of differential subordination, Filomat (2016), no. 14, 3743–3757.[37] D. K. Thomas and S. Verma, Invariance of the coefficients of strongly convex functions, Bull. Aust.Math. Soc. (2017), no. 3, 436–445.[38] D. G. Yang and J. L. Liu, A class of analytic functions with missing coefficients, Abstr. Appl. Anal. , Art. ID 456729, 16 pp.[39] Z. Ye, The logarithmic coefficients of close-to-convex functions, Bull. Inst. Math. Acad. Sin. (N.S.) (2008), no. 3, 445–452. Department of Mathematics, University of Delhi, Delhi–110 007, India
E-mail address : swati [email protected] Department of Mathematics, Aryabhatta College, Delhi-110021,India
E-mail address : [email protected] Bharati Vidyapeeth’s college of Engineering, Delhi-110063, India
E-mail address ::