FABER POLYNOMIAL COEFFICIENTS FOR GENERALIZED BI SUBORDINATE FUNCTIONS OF COMPLEX ORDER

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1 Journal of Mathematical Inequalities Volume, Number 3 (08), doi:0.753/jmi FABER POLYNOMIAL COEFFICIENTS FOR GENERALIZED BI SUBORDINATE FUNCTIONS OF COMPLEX ORDER ERHAN DENIZ, JAY M. JAHANGIRI, SAMANEH G. HAMIDI AND SIBEL K. KINA (Communicated by J. Pečarić) Abstract. In this paper, we obtain the upper bounds for the n-th (n 3) coefficients for generalized bi-subordinate functions of complex order by using Faber polynomial expansions. The results, which are presented in this paper, would generalize those in related works of several earlier authors.. Introduction Let A be the class of analytic functions in the open unit disk D = z C : z < } let S be the class of function f that are univalent in D are of the form f = z k= a k z k. A function f A is said to be subordinate to a function g A, denoted by f g, if there exists a function w A with w(0) = 0 w < satisfying f = g(w). We let S consist of starlike functions f A, that is Rez f f > 0 in D C consist of convex functions f A, that is Rez f f > 0 in D. In terms of subordination, these conditions are, respectively, equivalent to S f A : z f f z } z C f A : z f f z }. z A generalization of the above two classes, according to Ma Minda [0], are S z f } (ϕ) f A : f ϕ Mathematics subject classification (00): Primary 30C45, Secondary 30C80. Keywords phrases: Fabel polynomials, analytic, subordinate, convex, starlike, bi-univalent functions. c Ð, Zagreb Paper JMI

2 646 E. DENIZ, J. M. JAHANGIRI, S. G. HAMIDI AND S. K. KINA C(ϕ) f A : z f } f ϕ where ϕ is a positive real part function normalized by ϕ(0) =, ϕ (0) > 0 ϕ maps D onto a region starlike with respect to symmetric with respect to the real axis. Obvious extensions of the above two classes (see []) are S (γ;ϕ) f A : ( z f ) } γ f ϕ; γ C \ 0} C (γ;ϕ) f A : ( z f ) } γ f ϕ; γ C \ 0}. In literature, the functions belonging to these classes are called Ma-Minda starlike convex of complex order γ (γ C 0}), respectively. Some of the special cases of the above two classes S (γ;ϕ) C(γ;ϕ) are () S (,(Az) (Bz))= S [A,B] C (,(Az) (Bz))=C[A,B], ( B < A ) are classes of Janowski starlike convex functions, respectively, () S (( β)e iδ cosδ,(z) ( z))= S [δ,β] C(( β)e iδ cosδ, (z) ( z)) = C[δ,β], ( δ < π, 0 β < ) are classes of δ -spirallike δ -Robertson univalent functions of order β, respectively, (3) S (,(( β)z) ( z)) = S (β) C (,(( β)z)) ( z)) = C(β) (0 β < ) are classes of starlike convex functions of order β, respectively, ) β ) = S β C ( (4) S (, ( z z, ( z z convex functions of order β, respectively, (5) S (, z ) = SL = f A : ) β ) = C β are class of strongly starlike ( z f f ) < } is class of lemniscate starlike functions, (6) S (γ,(z) ( z))=s [γ] C(γ,(z) ( z))=c[γ] (γ C 0}) are classes of starlike convex functions of complex order, ) respectively, (7) S (,q k ) = k SP = f A : Re > k is class of ( z f f } z f f k-parabolik starlike functions, ( ) } (8) C (,q k ) = k U CV = f A : Re z f f > k z f f is class of k-uniformly convex functions. Here, for 0 k< the function q k : D w=uiv C : u > k ( (u ) v ), u > 0} has the form q k = Q zq z..., (z D) where B ; 0 k <, k 8 Q = ; k =, π π 4(k ) t(t)k (t) ; k >,, Q = (B ) 3 Q ; 0 k <, 3 Q ; k =, [4K (t)(t 6t) π ] 4 t(t)k (t) Q ; k >, (.)

3 FABER POLYNOMIAL COEFFICIENTS FOR GENERALIZED BI-SUBORDINATE FUNCTIONS 647 with B = π arccosk K (t) is the complete elliptic integral of first kind (see [8]). A function f A is said to be bi-univalent in D if both f its inverse map f are univalent in D. Let σ be the class of functions f S that are bi-univalent in D. For a brief history interesting examples of functions which are in (or are not in) the class σ, including various properties of such functions we refer the reader to the work of Srivastava et al. [] references therein. Bounds for the first few coefficients of various subclasses of bi-univalent functions were obtained by a variety of authors including ([4, 5, 6, 7], [0], [9], [3, 4, 5, 6, 7]). Not much was known about the bounds of the general coefficients a n ;n 4 of subclasses of σ up until the publication of the article [4] by Jahangiri Hamidi followed by a number of related publications (see [] [7]). In this paper, we apply the Faber polynomial expansions to certain subclasses of bi-univalent functions obtain bounds for their n th;(n 3) coefficients subject to a given gap series condition.. Coefficient estimates In the sequel, it is assumed that ϕ is an analytic function with positive real part in the unit disk D, satisfying ϕ(0) =, ϕ (0) > 0, ϕ(d) is symmetric with respect to the real axis. Such a function is known to be typically real with the series expansion ϕ = z B z B 3 z 3... > 0. Motivated by a class of functions defined by the first author [7], we define the following comprehensive class of analytic functions S (λ,γ;ϕ) f A : γ ( z f λ z f ) ( λ) fλ z f ϕ; 0 λ, γ C \ 0} }. A function f A is said the be generalized bi-subordinate of complex order γ type λ if both f its inverse map g = f are in S (λ,γ;ϕ). As special cases of the class S (λ,γ;ϕ) we have S (0,γ;ϕ) S (γ;ϕ) S (,γ;ϕ) C (γ;ϕ). In the following theorem we use the Faber polynomials introduced by Faber [9] to obtain a bound for the general coefficients of the bi-univalent functions in S (λ, γ; ϕ) subject to a gap series condition. THEOREM.. Let 0 λ γ C 0}. If both functions f = z n= ρ nz n its inverse map g = f are in S (λ,γ;ϕ) ρ m = 0; m n then γ ρ n (n )(λ(n )). Proof. If we write Λ( f) = λ z f ( λ) f then f S (λ,γ;ϕ) ( zλ ) ( f) γ Λ( f) ϕ

4 648 E. DENIZ, J. M. JAHANGIRI, S. G. HAMIDI AND S. K. KINA g = f S (λ,γ;ϕ) γ ( wλ ) (g(w)) Λ(g(w)) ϕ(w). We observe that a n = (λ(n ))ρ n for Λ( f) = z n= a nz n. Now, an application of Faber polynomial expansion to the power series S (λ,γ;ϕ) (e.g. see [] or [3, equation (.6)]) yields ( zλ ) ( f) γ Λ( f) = γ F n (a,a 3,a 4,...,a n )z n (.) where F n (a,a 3,...,a n ) = n= i i...(n )i n =n ( ) A(i,i,...,i n ) a i a i 3...a i n n A(i,i,...,i n ) := ( ) (n )i...ni n (i i... i n )!(n ). (i!)(i!)...(i n!) The first few terms of F n (a,a 3,...,a n ) are F = a, F = a a 3, F 3 = a 3 3a a 3 3a 4, F 4 = a 4 4a a 3 4a a 4 a 3 4a 5, F 5 = a 5 5a3 a 3 5a a 4 5 ( a 3 a 5) a 5a 3 a 4 5a 6. By the same token, the coefficients of the inverse map g = f may be expressed by where K n n = g(w) = f (w) = w n= n K n n (a,a 3,...,a n )w n = w n= τ n w n ( n)! ( n)! ( n)!(n )! an (( n))!(n 3)! an 3 a 3 ( n)! ( n)! [ ( n3)!(n 4)! an 4 a 4 (( n))!(n 5)! an 5 a5 ( n)a 3] ( n)! ( n5)!(n 6)! an 6 [a 6 ( n5)a 3 a 4 ] a n j V j V j for 7 j n is a homogeneous polynomial in the variables a 3,a 4,...,a n. Obviously, ( wλ ) (g(w)) γ Λ(g(w)) = γ F n (b,b 3,b 4,...,b n )w n (.) where b n = (λ(n ))τ n. Since, both functions f its inverse map g = f are in S (λ,γ;ϕ), by the definition of subordination, there exist two Schwarz functions n= j 7

5 FABER POLYNOMIAL COEFFICIENTS FOR GENERALIZED BI-SUBORDINATE FUNCTIONS 649 u = c zc z...c n z n..., u <, z D v(w) = d wd w... d n w n..., v(w) <, w D, so that ( zλ ) ( f) γ Λ( f) = ϕ(u) = Kn (c,c,...,c n,,b,...,b n )z n n= (.3) ( wλ ) (g(w)) γ Λ(g(w)) = ϕ(v(w)) = Kn (d,d,...,d n,,b,...,b n )w n. n= (.4) In general (e.g., see [] [, equation (.6)]), the coefficients Kn p := Kn p (k,k,...,k n,,b,...,b n ) are given by Kn p p! B n p! B n = (p n)!n! kn (p n)!(n )! kn k p! B n (p n)!(n 4)! kn 3 k 3 [ p! B n 3 (p n3)!(n 4)! kn 4 k 4 p n3 k p! (p n4)!(n 5)! kn 5 ] B n [ B n 4 B n 3 k 5 (p n4)k k 3 ] k n j X j j 6 where X j is a homogeneous polynomial of degree j in the variables k,k 3,...,k n. For the coefficients of the Schwarz functions u v(w) we have c n d n (e.g., see [8]). Comparing the corresponding coefficients of (.) (.3) yields γ F n (a,a 3,...,a n ) = Kn (c,c,...,c n,,b,...,b n ) (.5) which under the assumption a m = 0; m n we get γ (n )a n = γ (n )(λ(n ))ρ n = c n. (.6) Similarly, comparing the corresponding coefficients of (.) (.4) gives γ F n (b,b 3,...,b n ) = K n (d,d,...,d n,,b,...,b n ) (.7) which by the hypothesis, we obtain γ (n )b n = d n. Note that, for a m = 0; m n we have b n = a n therefore γ (n )a n = γ (n )(λ(n ))ρ n = d n. (.8)

6 650 E. DENIZ, J. M. JAHANGIRI, S. G. HAMIDI AND S. K. KINA Taking the absolute values of either of the equations (.6) or (.8) we obtain the required bound. To prove our next theorem, we shall need the following well-known lemma (see [8]). LEMMA.. ([8]) Let the function p= n= p nz n be so that Re (p) > 0 for z D. Then for < α <, p α p α p ; α <, ( α) p ; α. (.9) Let ϕ = n= ϕ nz n be a Schwarz function so that ϕ <, z D. Set p = [ ϕ ]/[ ϕ ] where p = n= p nz n is so that Re (p) > 0 for z D. Comparing the corresponding coefficients of powers of z yields p = ϕ p = ( ϕ ϕ). Now, substituting for p p letting η = α in (.9) we obtain ϕ ηϕ ( η) ϕ ; η > 0, (η) ϕ ; η 0. (.0) The following theorem gives bounds for the coefficient body (ρ,ρ 3 ) of the generalized bi-subordinate functions of complex order γ type λ. THEOREM.. Let 0 λ γ C 0}. If both functions f = z n= ρ nz n its inverse map g = f are in S (λ,γ;ϕ) then ρ 3 ρ Proof. For n =, (.5) (.7) imply γ (λ) ; B, γb (λ) ; < B ρ = γc λ ρ = γd λ. (.) For n = 3, the equations (.5) (.7), respectively, imply. (λ)ρ 3 (λ) ρ γ = c B c (.) (λ)ρ 3 (36λ λ )ρ γ = d B d. (.3) Considering (.) we get c = d. Also, from (.), (.3) > 0 we find that [( ρ 3 ρ = γ c B ) ( c d B )] d. (.4) 4(λ)

7 FABER POLYNOMIAL COEFFICIENTS FOR GENERALIZED BI-SUBORDINATE FUNCTIONS 65 Taking the absolute values of both sides of (.4) gives [ ρ 3 ρ γ c B c 4(λ) d B ] d. (.5) If B 0, then for η = B / apply (.0) to (.5) to get [ ( ] [ ( ]} ρ 3 ρ γ B B ) c B B ) d. (.6) 4(λ) yields If B > 0 then (.6) yields ρ 3 ρ γ (λ). If B < 0 then for the maximum values c = d = the inequality (.6) ρ 3 ρ γ B [ 4(λ) ( B B )]} = γ B (λ). If B > 0, then for η = B / apply (.0) to (.5) to get ρ 3 ρ γ B [ ( ] [ ( ]} B B ) c B B ) d. (.7) 4(λ) If B > 0 then (.7) yields ρ 3 αρ γ (λ). If B < 0 then for the maximum values c = d = the inequality (.7) yields ρ 3 ρ γ B [ 4(λ) This concludes the proof of Theorem.. ( B B )]} = γ B (λ). For different values of λ γ, Theorems.. yield the following interesting corollaries. COROLLARY.3. If both functions f its inverse map g = f are in S (γ;ϕ), then ρ n γ (n ), ρ m = 0; m n. Taking ϕ = (Az) (Bz) = (A B)z B(A B)z... in Corollary.3, we obtain the result of Hamidi Jahangiri (see [3]). COROLLARY.4. If both functions f its inverse map g = f are in C (γ;ϕ), then ρ n γ (n )n, ρ m = 0; m n. COROLLARY.5. If both functions f its inverse map g = f are in S [δ,β] C [δ,β], respectively, then ρ n ( β) cosδ (n ) ρ n ( β) cosδ, ρ m = 0; m n. n(n )

8 65 E. DENIZ, J. M. JAHANGIRI, S. G. HAMIDI AND S. K. KINA COROLLARY.6. If both functions f its inverse map g = f are in k SP k U CV, respectively, then for ρ m = 0; m n we have B ρ n Q (n ), ρ 3 ρ ; 0 k <, k 4 ; k =, π π 8(k ) t(t)k (t) ; k > B ρ n Q (n )n, ρ 3 ρ ; 0 k <, 3( k ) 4 ; k =, 3π π 4(k ) t(t)k (t) ; k > where Q is given by (.). Proof. Let f its inverse map g = f be in k SP. We will show that Q Q for k 0. First suppose 0 k <. Since 0 arccosk π we have Q Q = B 3. For k = it is clear that Q Q = 3 <. Finally, for k > we get Q Q = [4K (t)(t 6t) π ] 4. A similar argument can be used to justify the case for t(t)k (t) k U C V. Determination of extremal functions for bi-univalent functions (in general) for bi-subordinate functions (in particular) remains a challenge. R E F E R E N C E S [] H. AIRAULT, Remarks on Faber polynomials, Int. Math. Forum 3 (9 ) (008) , MR [] H. AIRAULT, A. BOUALI, Differential calculus on the Faber polynomials, Bull. Sci. Math. 30 (3) (006) 79, MR5663. [3] H. AIRAULT, J. REN, An algebra of differential operators generating functions on the set of univalent functions, Bull. Sci. Math. 6 (5) (00) , MR9475. [4] R. M. ALI, S. K. LEE, V. RAVICHANDRAN, S. SUPRAMANIAM, Coefficient estimates for biunivalent Ma Minda starlike convex functions, Appl. Math. Lett. 5 (3) (0) , MR [5] S. BULUT, Faber polynomial coefficient estimates for a comprehensive subclass of analytic biunivalent functions, C. R. Acad. Sci. Paris, Ser. I 35 (6) (04) , MR308. [6] M. ÇAĞLAR, H. ORHAN AND N. YAĞMUR, Coefficient bounds for new subclasses of bi-univalent functions, Filomat 7 (7) (03), 65-7, MR [7] E. DENIZ, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Class. Anal. () (03) 49 60, MR334. [8] P. L. DUREN, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, vol. 59, Springer, New York, 983, MR [9] G. FABER, Über polynomische Entwickelungen, Math. Ann. 57 (3) (903) , MR56. [0] B. A. FRASIN, M. K. AOUF, New subclasses of bi-univalent functions, Appl. Math. Lett. 4 (0) , MR8037. [] S. G. HAMIDI, J. M. JAHANGIRI, Faber polynomial coefficient estimates for analytic bi-close-toconvex functions, C. R. Acad. Sci. Paris, Ser. I 35 () (04) 7 0, MR [] S. G. HAMIDI, J. M. JAHANGIRI, Faber polynomial coefficient estimates for bi-univalent functions defined by subordinations, Bull. Iran. Math. Soc. 4 (5) (05) 03 9, MR34668.

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