Coefficient bounds for some subclasses of p-valently starlike functions

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1 doi: /v y ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN POLONIA VOL. LXVII, NO. 2, 203 SECTIO A C. SELVARAJ, O. S. BABU G. MURUGUSUNDARAMOORTHY Coefficient bounds for some subclasses of p-valently starlike functions Abstract. For functions of the form f(z) =z p + n= ap+nzp+n we obtain sharp bounds for some coefficients functionals in certain subclasses of starlike functions. Certain applications of our main results are also given. In particular, Fekete Szegö-like inequality for classes of functions defined through extended fractional differintegrals are obtained.. Introduction. Let A p denote the class of all functions of the form (.) f(z) =z p + a p+n z p+n (p N =, 2, 3,...}) n= which are analytic in the open disk = z C : z < } let A = A. For f(z) given by (.) g(z) given by g(z) =z p + n= b p+nz p+n, their convolution (or Hadamard product), denoted by f g, is defined as (f g)(z) =z p + a p+n b p+n z p+n. n= Given two functions f g, which are analytic in, the function f is said to be subordinate to g in, written f g or f(z) g(z), if there exists a Schwarz function w(z), analytic in with w(0) = 0 w(z) < such that f(z) =g(w(z)), z. In particular, if the function g is univalent in, the above subordination is equivalent to f(0) = g(0) f( ) g( ) Mathematics Subject Classification. 30C45. Key words phrases. Analytic functions, starlike functions, convex functions, p- valent functions, subordination, convolution, Fekete Szegö inequality.

2 66 C. Selvaraj, O. S. Babu G. Murugusundaramoorthy Definition.. Let φ(z) be an analytic function with positive real part in the unit disk with φ(0) = φ (0) > 0 that maps onto a region starlike with respect to symmetric with respect to the real axis. The class M p (λ; φ) is the subclass of A p consisting of functions f(z) satisfying (.2) p zf (z) ( λ)z p φ(z) (z, 0 <λ ). + λf(z) As special cases, let M p (; φ) =S p(φ) = f(z) A p : zf } (z) p f(z) φ(z), } M (; φ) =S (φ) = f(z) A: zf (z) f(z) φ(z). When φ(z) = +Az +Bz, B<A, we denote the subclass M p(λ; φ) by M p (λ; A, B). The class M (; A, B) =S [A, B] was studied by Janowski [2]. For 0 α<, let M p (λ; α) =M p (λ; 2α, ). For a fixed analytic function g A p with positive coefficients, define the class M p,g (λ; φ) as the class of all functions f A p satisfying f g M p (λ; φ). This class includes as special cases several other classes studied in the literature. For example, when g(z) =z p + p+n n= p zp+n, the class M p,g (; φ) reduces to the class C p ( : φ) =C p (φ) consisting of functions f A p satisfying ( ) (.3) + zf (z) p f φ(z), z. (z) The classes S (φ) C(φ) =C (φ) were introduced studied by Ma Minda [4]. They have obtained the Fekete Szegö inequality for functions in the class C(φ). Since f(z) C(φ) if only if zf (z) S (φ), we get the Fekete Szegö inequality for functions in the class S (φ). For a brief history of Fekete Szegö problem for classes of starlike, convex close-to-convex functions see the recent paper by Srivastava et al. [0]. Let Ω be the class of analytic functions of the form (.4) w(z) =w z + w 2 z in the unit disk satisfying the condition w(z) <. There has been triggering interest in the literature (see [, 3, 4, 9, 0]) to define certain subclasses of analytic functions to discuss Fekete Szegö inequalities. Making use of the techniques, in this paper we defined two new classes M p (λ; φ) M p,g (λ; φ) to obtain Fekete Szegö inequalities to discuss the results on upper bounds for the coefficient a p+3.

3 Coefficient bounds for some subclasses of p-valently starlike functions 67 Lemma.2 ([]). If w Ω, then t if t, w 2 tw 2 if t, t if t. When t< or t>, equality holds if only if w(z) =z or one of its rotations. If <t<, then equality holds if only if w(z) =z 2 or one of its rotations. Equality holds for t = if only if w(z) = z(λ+z) +λz, (0 λ ) or one of its rotations while for t =, equality holds if only if w(z) = z(λ+z) +λz, (0 λ ) or one of its rotations. Also sharp upper bound above can be improved as follows when < t<: w 2 tw 2 +(+t) w 2 ( <t 0) w 2 tw 2 +( t) w 2 (0 <t<). Lemma.2 is a reformulation of Lemma of Ma Minda [4]. Lemma.3 ([3]). If w Ω, then for any complex number t w 2 tw max, 2 t }. The result is sharp for the functions w(z) =z or w(z) =z 2. Lemma.4 ([8]). If w Ω, then for any real numbers q q 2 the following sharp estimate holds: w 3 + q w w 2 + q 2 w H(q 3,q 2 ) where for (q,q 2 ) D D 2 q 2 for (q,q 2 ) 7 D k k=3 ( ) 2 H(q,q 2 )= 3 ( q q + 2 +) for (q,q 2 ) D 8 D 9 3( q ++q 2 ) ( )( ) q 2 q 2 4 q q 2 4q for (q,q 2 ) D 0 D \±2,} 2 3(q 2 ) ( ) 2 3 ( q q 2 ) for (q,q 2 ) D 2. 3( q q 2 ) The extremal functions, up to rotations, are of the form w(z) =z 3, w(z) =z, w(z) =w 0 (z) = (z[( λ)ε 2 + λε ] ε ε 2 z), [( λ)ε + λε 2 ]z w(z) =w (z) = z(t z) t z, w(z) =w 2(z) = z(t 2 + z) +t 2 z,

4 68 C. Selvaraj, O. S. Babu G. Murugusundaramoorthy ε = ε 2 =, ε = t 0 e iθ 0 2 (a b), ε2 = e iθ 0 2 (ia ± b), t 0 = a = t 0 cos θ 0 2, b = t 2 θ 0 sin2 0 2, λ = b ± a 2b, [ 2q2 (q 2 +2) ] ( 3q2 2, 3(q 2 )(q 2 4q t = 2) q + 3( q ++q 2 ) ) 2, ( ) q 2 θ 0 t 2 =, cos 3( q q 2 ) 2 = q [ q2 (q 2 +8) 2(q2 +2) ] 2 2q 2 (q 2 +2). 3q2 The sets D k, k =, 2,...,2 are defined as follows: D = (q,q 2 ): q } 2, q 2, D 2 = (q,q 2 ): 2 q 2, 4 } 27 ( q +) 3 ( q +) q 2, D 3 = (q,q 2 ): q } 2,q 2, D 4 = (q,q 2 ): q 2,q 2 2 } 3 ( q +), D 5 =(q,q 2 ): q 2,q 2 }, D 6 = (q,q 2 ):2 q 4,q 2 } 2 (q2 +8), D 7 = (q,q 2 ): q 4,q 2 2 } 3 ( q ), D 8 = (q,q 2 ): 2 q 2, 2 3 ( q +) q 2 4 } 27 ( q +) 3 ( q +), D 9 = (q,q 2 ): q 2, 2 3 ( q +) q 2 2 q } ( q +) q 2 +2 q, +4 D 0 = (q,q 2 ):2 q 4, 2 q ( q +) q 2 +2 q +4 q 2 } 2 (q2 +8), D = (q,q 2 ): q 4, 2 q ( q +) q 2 +2 q +4 q 2 2 q ( q ) q 2 2 q +4 D 2 = (q,q 2 ): q 4, 2 q ( q ) q 2 2 q +4 q 2 2 } 3 ( q ). },

5 Coefficient bounds for some subclasses of p-valently starlike functions Coefficient bounds. By making use of Lemmas.2.4, we prove the following bounds for the class M p (λ; φ). Theorem 2.. Let φ(z) =+B z + B 2 z , where B n s are real with B > 0 B 2 0, let 0 <λ, σ := [pb2 λ +(B 2 B )(p pλ +)](p pλ +) ()pb 2, σ 2 := [pb2 λ +(B 2 + B )(p pλ +)](p pλ +) ()pb 2, σ 3 := [pb2 λ + B 2(p pλ +)](p pλ +) ()pb 2. If f(z) given by (.) belongs to M p (λ; φ), then Λ if μ σ (2.) a p+2 μa 2 p+ if σ μ σ 2 Λ if μ σ 2. Further, if σ μ σ 3, then (2.2) a p+2 μa 2 p+ + If σ 3 μ σ 2, then (2.3) a p+2 μa 2 p+ + where (p pλ +)2 () ( + Λ) a p+ 2 (p pλ +)2 () ( Λ) a p+ 2., Λ= μ()pb2 λ(p pλ +)pb2 (p pλ +)2 B 2 (p pλ +) 2. B For any complex number μ, (2.4) a p+2 μa 2 p+ Further, max, Λ } (2.5) a p+3 p pλ +3 H(q,q 2 ), where H(q,q 2 ) is as defined in Lemma.3, q := 2B 2 B (2pλ 2p 3)λ (p pλ +)()

6 70 C. Selvaraj, O. S. Babu G. Murugusundaramoorthy q 2 := B 3 (2pλ 2p 3)λpB 2 λ 2 p 2 B 2. B (p pλ +)(p pλ +2) These results are sharp. Proof. If f(z) M p (λ; φ), then there is an analytic function w(z) given by (.4) such that p (2.6) zf (z) ( λ)z p + λf(z) = φ(w(z)). Since p zf (z) ( λ)z p + λf(z) =+ [ ] [( (p +) λ a p+ z + p p + (λ 2 λp ) ] [( (p +) a 2 p+ z (2λ 2 λp (p +2) λp ) (p +) a p+ a p+2 + ( ) ] p (p +)λ2 λ 3 a 3 p+ z ) (p +2) λ (p +3) λ p a p+2 ) a p+3 φ(w(z)) = +B w z+(b w 2 +B 2 w)z 2 2 +(B w 3 +2B 2 w w 2 +B 3 w)z , we have from (2.6), (2.7) (2.8) (2.9) Now, a p+ = w p pλ +, a p+2 = p(b w 2 + B 2 w 2 ) a p+3 = p pλ [ 2B2 w 3 + λp 2 B 2 w2 (p pλ +)() (2pλ 2p 3)λ B (p pλ +)(p pλ +2) ] } w 3. [ B3 (2pλ 2p 3)λpB 2 λ 2 p 2 B 2 B (p pλ +)(p pλ +2) ] w w 2 (2.0) a p+2 μa 2 p+ = w 2 Λw}. 2 The results (2.) (2.3) are established by an application of Lemma.2, inequality (2.4) by Lemma.3 (2.5) follows from Lemma.4. To show

7 Coefficient bounds for some subclasses of p-valently starlike functions 7 that the bounds in (2.), (2.2) (2.3) are sharp, we define K φn, n = 2, 3, 4,... by p zk φn (z) ( λ)z p + λk φn (z) = φ(zn ), K φn (0) = 0 = K φn (0) the functions F λ G λ, 0 <λ by p zf λ (z) ( λ)z p + λf λ (z) = φ p zg λ (z) ( λ)z p + λg λ (z) = φ ( z(z + λ) +λz ( ), F λ (0) = 0 = F λ (0) ) z(z + λ), G λ (0) = 0 = G λ (0). +λz Clearly the functions K φn, F λ, G λ M p (λ; φ). If μ<σ or μ>σ 2, then equality holds if only if f is K φ2 or one of its rotations. When σ <μ<σ 2, equality holds if only if f is K φ3 or one of its rotations. If μ = σ then equality holds if only if f is F λ or one of its rotations. Equality holds for μ = σ 2 if only if f is G λ or one of its rotations. Remark 2.2. For λ =, results (2.) (2.4) coincide with the results obtained for the class S p(φ) by Ali et al. []. Remark 2.3. For λ =, p =, results (2.) (2.4) coincide with the results obtained for the class S (φ) by Ma Minda [4]. Example 2.4. Let B<A. If f(z) given by (.) belongs to M p (λ; A, B), then p(a B)Λ if μ σ a p+2 μa 2 p(a B) p+ if σ μ σ 2 p(a B)Λ if μ σ 2. Further, if σ μ σ 3, then a p+2 μa 2 p+ + If σ 3 μ σ 2, then a p+2 μa 2 p+ + (p pλ +) 2 p(a B)() ( + Λ) a p+ 2 (p pλ +) 2 p(a B)() ( Λ) a p+ 2 p(a B). p(a B),

8 72 C. Selvaraj, O. S. Babu G. Murugusundaramoorthy where [p(a B)λ ( + B)(p pλ +)](p pλ +) σ :=, p()(a B) [p(a B)λ +( B)(p pλ +)](p pλ +) σ 2 :=, p()(a B) [p(a B)λ B(p pλ +)](p pλ +) σ 3 := p()(a B) μp()(a B) (p pλ +)[(A B)λp (p pλ +)B] Λ A = (p pλ +) 2. For any complex number μ, a p+2 μa 2 p(a B) p+ max, Λ A }. In particular, if f M p (λ; α), then 2p( α)λ a p+2 μa 2 2p( α) p+ 2p( α)λ Further, if σ μ σ 3, then a p+2 μa 2 p+ + If σ 3 μ σ 2, then where a p+2 μa 2 p+ + Λ α = if μ σ if σ μ σ 2 if μ σ 2. (p pλ +) 2 2p( α)() ( + Λ) a p+ 2 (p pλ +) 2 2p( α)() ( Λ) a p+ 2 λ(p pλ +) σ :=, (p pλα +)(p pλ +) σ 2 :=, p( α)() [2p( α)λ +(p pλ +)](p pλ +) σ 3 := 2p( α)() 2p( α). 2p( α), 2μp( α)() (p pλ +)[2λp( α)+(p pλ +)] (p pλ +) 2.

9 Coefficient bounds for some subclasses of p-valently starlike functions 73 For any complex number μ, The results are sharp. a p+2 μa 2 p+ 2p( α) max, Λ α }. Corollary 2.5. Let φ(z) be as in Theorem 2., g(z) =z p + g p+n z p+n (g p+n > 0), let n= σ := g2 p+ [pb 2λ +(B 2 B )(p pλ +)](p pλ +) g p+2 ()pb 2, σ 2 := g2 p+ [pb 2λ +(B 2 + B )(p pλ +)](p pλ +) g p+2 ()pb 2, σ 3 := g2 p+ [pb 2λ + B 2(p pλ +)](p pλ +) g p+2 ()pb 2. If f(z) given by (.) belongs to M p,g (λ; φ), then Λ if μ σ g p+2 () (2.) a p+2 μa 2 p+ if σ μ σ 2 g p+2 () pb Λ if μ σ 2. g p+2 () Further, if σ μ σ 3, then (2.2) If σ 3 μ σ 2, then (2.3) where Λ g = a p+2 μa 2 p+ + g2 p+ (p pλ +)2 ( + Λ) a p+ 2 g p+2 () g p+2 (). a p+2 μa 2 p+ + g2 p+ (p pλ +)2 ( Λ) a p+ 2 g p+2 () g p+2 (), g p+2 g 2 p+ μ()pb 2 λ(p pλ +)pb2 (p pλ +)2 B 2 (p pλ +) 2 B.

10 74 C. Selvaraj, O. S. Babu G. Murugusundaramoorthy For any complex number μ, a p+2 μa 2 p+ g p+2 () max, Λ g }. Further, a p+3 g p+3 (p pλ +3) H(q,q 2 ), where H(q,q 2 ) is as defined in Lemma.3, q := 2B 2 B (2pλ 2p 3)λ (p pλ +)() q 2 := B 3 (2pλ 2p 3)λpB 2 λ 2 p 2 B 2. B (p pλ +)(p pλ +2) These results are sharp. 3. Applications to functions defined by extended fractional differintegrals. With a view to define fractional differintegral operator Ω (δ,p) z, we recall Gauss hypergeometric function 2 F defined by [6]. (3.) 2F (a, b; c; z) = n=0 (a) n (b) n (c) n z n n! (a, b, c C, c 0,, 2,...), where (d) n denotes the Pochhammer symbol given in terms of Gamma function Γ by Γ(d + n) (n =0; d C \0}) (d) n = = Γ(d) d(d +)...(d + n ) (n N; d C). We note that the series defined by (3.) converges absolutely for z hence 2 F represents an analytic function in. Also we recall the definitions of fractional calculus considered by Owa [5] (see also [6,, 2]). Definition 3.. The fractional integral of order δ (δ > 0) is defined, for a function f, analytic in a simply connected region of the complex plane containing the origin, by (3.2) Dz δ f(z) = z Γ(δ) 0 f(ζ) dζ, (z ζ) δ where the multiplicity of (z ζ) δ is removed by requiring log(z ζ) to be real when z ζ>0.

11 Coefficient bounds for some subclasses of p-valently starlike functions 75 Definition 3.2. Under the hypothesis of Definition 3., the fractional derivative of f of order δ (δ 0) is defined by d z f(ζ) dζ, (0 δ<) (3.3) Dzf(z)= δ Γ( δ) dz (z ζ) δ 0 d n dz n Dδ n z f(z), (n δ<n+;n N 0 ) where N 0 = N 0} the multiplicity of (z ζ) δ is removed as in Definition 3.. Definition 3.3 ([7]). The extended fractional differintegral operator Ω z (δ,p) : A p A p for a function f of the form (.) for a real number δ ( < δ<p+)isdefinedby (3.4) Ω (δ,p) z f(z) = Γ(p + δ) z δ D δ Γ(p +) zf(z) ( <δ<p+; z ), where D δ zf(z) is respectively, the fractional integral of f of order δ when <δ<0 the fractional derivative of f of order δ when 0 δ<p+. We note that Ω (δ,p) z f(z) =z p + n= Γ(n + p +)Γ(p + δ) Γ(p +)Γ(n + p + δ) a n+pz n+p = z p 2F (,p+;p + δ; z) f(z) ( <δ<p+; z ). Let M p,δ (λ; φ) be the class of functions f A p for which Ω (δ,p) z f(z) M p (λ; φ). The class M p,δ (λ; φ) is the special case of the class M p,g (λ; φ), when g(z) =z p Γ(n + p +)Γ(p + δ) + Γ(p +)Γ(n + p + δ) zn+p. n= Since ( ) Ω (δ,p) z f (z) =z p Γ(n + p +)Γ(p + δ) + Γ(p +)Γ(n + p + δ) a n+pz n+p, n= we have (3.5) Γ(p +2)Γ(p + δ) g p+ = Γ(p +)Γ(p +2 δ) = p + p + δ, (3.6) Γ(p +3)Γ(p + δ) g p+2 = Γ(p +)Γ(p +3 δ) = (p +)(p +2) (p + δ)(p +2 δ), (3.7) g p+3 = Γ(p +4)Γ(p + δ) Γ(p +)Γ(p +4 δ) = (p +)(p +2)(p +3) (p + δ)(p +2 δ)(p +3 δ).

12 76 C. Selvaraj, O. S. Babu G. Murugusundaramoorthy For g p+,g p+2 g p+3 given by (3.5), (3.6) (3.7), Corollary 2.5 reduces to the following: Theorem 3.4. Let φ(z) be as in Theorem 2., let σ := σ 2 := σ 3 := (p +)(p +2 δ) (p +2)(p + δ) (p +)(p +2 δ) (p +2)(p + δ) (p +)(p +2 δ) (p +2)(p + δ) [pb 2λ +(B 2 B )(p pλ +)](p pλ +) ()pb 2, [pb 2λ +(B 2 + B )(p pλ +)](p pλ +) ()pb 2, [pb 2λ + B 2(p pλ +)](p pλ +) ()pb 2. If f(z) given by (.) belongs to M p,δ (λ; φ), then (p + δ)(p +2 δ) Λ (p +)(p +2) () if μ σ, a p+2 μa 2 (p + δ)(p +2 δ) p+ if σ μ σ 2, (p +)(p +2) () (p + δ)(p +2 δ) Λ if μ σ 2. (p +)(p +2) () Further, if σ μ σ 3, then a p+2 μa 2 (p +)(p +2 δ) (p pλ +) 2 p+ + (p +2)(p + δ) () ( + Λ) a p+ 2 (3.8) (p + δ)(p +2 δ) (p +)(p +2) (). If σ 3 μ σ 2, then a p+2 μa 2 (p +)(p +2 δ) (p pλ +) 2 p+ + (p +2)(p + δ) () ( + Λ) a p+ 2 (3.9) (p + δ)(p +2 δ) (p +)(p +2) (), where (p+2)(p+ δ) (p+)(p+2 δ) Λ δ = μ()pb2 λ(p pλ +)pb2 (p pλ +)2 B 2 (p pλ +) 2. B For any complex number μ, Further, a p+2 μa 2 p+ a p+3 (p + δ)(p +2 δ) (p +)(p +2) (p + δ)(p +2 δ)(p +3 δ) (p +)(p +2)(p +3) () max, Λ δ }. (p pλ +3) H(q,q 2 ),

13 Coefficient bounds for some subclasses of p-valently starlike functions 77 where q := 2B 2 B (2pλ 2p 3)λ (p pλ +)() q 2 := B 3 (2pλ 2p 3)λpB 2 λ 2 p 2 B 2. B (p pλ +)(p pλ +2) These results are sharp. Acknowledgements. The authors thank the referee for his insightful suggestions. References [] Ali, R. M., Ravichran, V., Seenivasagan, N., Coefficient bounds for p-valent functions, Appl. Math. Comput. 87 (2007), [2] Janowski, W., Some extremal problems for certain families of analytic functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 2 (973), [3] Keogh, F. R., Merkes, E. P., A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (969), 8 2. [4] Ma, W. C., Minda, D., A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 992), Int. Press, Cambridge, MA, 994, [5] Owa, S., On the distortion theorem. I, Kyungpook Math. J. 8 () (978), [6] Owa, S., Srivastava, H. M., Univalent starlike generalized hypergeometric functions, Canad. J. Math. 39 (5) (987), [7] Patel, J., Mishra, A., On certain subclasses of multivalent functions associated with an extended differintegral operator, J. Math. Anal. Appl. 332 (2007), [8] Prokhorov, D. V., Szynal, J., Inverse coefficients for (α, β)-convex functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 35 (98), [9] Selvaraj, C., Selvakumaran, K. A., Fekete Szegö problem for some subclass of analytic functions, Far East J. Math. Sci. (FJMS) 29 (3) (2008), [0] Srivastava, H. M., Mishra, A. K., Das, M. K., The Fekete Szegö problem for a subclass of close-to-convex functions, Complex Variables Theory Appl. 44 (2) (200), [] Srivastava, H. M., Owa, S., An application of the fractional derivative, Math. Japon. 29 (3) (984), [2] Srivastava, H. M., Owa, S., Univalent Functions, Fractional Calculus their Applications, Halsted Press/John Wiley & Sons, Chichester New York, 989. C. Selvaraj O. S. Babu Department of Mathematics Department of Mathematics Presidency College (Autonomous) Dr. Ambedkar Govt. Arts College Chennai Chennai India India pamc9439@yahoo.co.in osbabu009@gmail.com

14 78 C. Selvaraj, O. S. Babu G. Murugusundaramoorthy G. Murugusundaramoorthy School of Advanced Sciences VIT university Vellore India gmsmoorthy@yahoo.com Received June 25, 202