MSC 2010: 30C45The universally prestarlike functions of order α ≤ 1 in the slit domain
Λ = C [1;∞) have been recently introduced by S. Ruscheweyh. This notion
generalizes the corresponding one for functions in the unit disk Δ (and other
circular domains in C). In this paper, we obtain the coefficient inequalities
and the Fekete-Szegö inequality for such functions

Lourthu Mary, J.

Shanmugam, T.

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

FEKETE-SZEGO¨ INEQUALITY FORUNIVERSALLY PRESTARLIKE FUNCTIONST.N. Shanmugam ∗, J. Lourthu Mary ∗∗This paper is dedicated to the 70th anniversary of Professor SrivastavaAbstractThe universally prestarlike functions of order α ≤ 1 in the slit domainΛ = C\[1,∞) have been recently introduced by S. Ruscheweyh. This notiongeneralizes the corresponding one for functions in the unit disk ∆(and othercircular domains in C). In this paper, we obtain the coefficient inequalitiesand the Fekete-Szego¨ inequality for such functions.MSC 2010: 30C45Key Words and Phrases: prestarlike functions, universally prestarlikefunctions, coefficients, Fekete-Szego¨ inequality1. IntroductionLet H(Ω) denote the set of all analytic functions defined in a domainΩ. For domain Ω containing the origin H0(Ω) stands for the set of allfunction f ∈ H(Ω) with f(0) = 1. We also use the notation H1(Ω) ={zf : f ∈ H0(Ω)} . In the special case when Ω is the open unit disk ∆ ={z ∈ C : |z| < 1} , we use the abbreviation H,H0 and H1 respectively forH(Ω),H0(Ω) and H1(Ω).c© 2010, FCAA – Diogenes Co. (Bulgaria). All rights reserved.386 T.N. Shanmugam , J. Lourthu MaryA function f ∈ H1 is called starlike of order α with (0 ≤ α < 1) satisfyingthe inequality<{zf ′(z)f(z)}> α (z ∈ ∆) (1.1)and the set of all such functions is denoted by Sα. The convolution orHadamard Product of two functions f(z) =∞∑n=0anzn and g(z) =∞∑n=0bnznis defined as(f ∗ g)(z) =∞∑n=0anbnzn.A function f ∈ H1 is called prestarlike of order α ifz(1− z)2−2α ∗ f(z) ∈ Sα. (1.2)The set of all such functions is denoted by Rα. The notion of prestarlikefunctions has been extended from the unit disk to other disk and half planescontaining the origin. Let Ω be one such disk or half plane.Then there aretwo unique parameters γ ∈ C \ {0} and ρ ∈ [0, 1] such thatΩγ,ρ = {wγ,ρ(z) : z ∈ ∆} , (1.3)wherewγ,ρ(z) =γz1− ρz .Note that 1 /∈ Ωγ,ρ iff |γ + ρ| ≤ 1.Definition 1.1. (see [2], [3], [4]) Let α ≤ 1, and Ω = Ωγ,ρ for someadmissible pair (γ, ρ). A function f ∈ H1(Ωγ,ρ) is called prestarlike of orderα in Ωγ,ρ iffγ,ρ(z) =1γf(wγ,ρ(z)) ∈ Rα. (1.4)The set of all such functions f is denoted by Rα(Ω).Let Λ be the slit domain C \ [1,∞)(the slit being along the positive realaxis).Definition 1.2. (see [2], [3], [4]) Let α ≤ 1. A function f ∈ H1(Λ) iscalled universally prestarlike of order α if and only if f is prestarlike of orderα in all sets Ωγ,ρ with |γ + ρ| ≤ 1. The set of all such functions is denotedby Ruα.FEKETE-SZEGO¨ INEQUALITY FOR . . . 387Example 1.1. A function f(z) =z(1− z)1−2α is prestarlike of order0 ≤ α < 1. When α = 0 the function is universally prestarlike of order0.When α = 12 the function f(z) = z is the only entire function in Ruα.Example 1.2. A function f(z) =z(1− z) 12is universally prestarlike oforder 12 .Definition 1.3. (see [4]) Let φ(z) be an analytic function with positivereal part on ∆, which satisfies φ(0) = 1, φ′(0) > 0 and which maps the unitdisc ∆ onto a region starlike with respect to 1 and symmetric with respectto the real axis. Then the class Ruα(φ) consists of all analytic functionf ∈ H1(Λ) satisfyingD3−2αfD2−2αf≺ φ(z), (1.5)where ≺ denotes the subordination,where (Dβf)(z) = z(1−z)β ? f, for β ≥ 0.In particular, for β = n ∈ N. We have Dn+1f = zn!(zn−1f)(n). We letRuα(A,B) denote the class Ruα(φ), where φ(z) =1 +Az1 +Bz(−1 ≤ B < A ≤ 1).For suitable choices of A,B,α the class Ruα(A,B) reduces to several wellknown classes of functions. Ru12(1,−1) is the class S∗ of starlike univalentfunctions.Note 1.1. (see [4]) Let F (z) =∞∑k=0akzk =∫ 10dµ(t)1− tz where, ak =∫ 10tkdµ(t), µ(t) is a probability measure on [0, 1]. Let T denote the set ofall such functions F . They are analytic in the slit domain Λ.Note 1.2. (see [3]) Let Ω be a circular domain containing the origin,α ≤ 1, and let f ∈ Rα(Ω), F ∈ Ruα. Then f ∗ F ∈ Rα(Ω).To prove our result we need the following theorem.Theorem 1.1. (see [2], [4]) Let 0 ≤ α ≤ 1 and f ∈ H1(Λ). Then f ∈ Ruαif and only ifD3−2αfD2−2αf∈ T. (1.6)This admits an explicit representation of the function in Ruα. If f ∈ H0 hasall its Taylor coefficients at the origin different from zero we write f (−1) forthe (possibly formal but) unique solution of f ∗ f (−1) = 11−z .388 T.N. Shanmugam , J. Lourthu MaryLemma 1.1. (see [1]) If P1(z) = 1 + c1z + c2z2 + . . . is an analyticfunction with positive real part in ∆, then|c2 − vc21| ≤−4v + 2, v ≤ 02, 0 ≤ v ≤ 14v + 2, v ≥ 1when v < 0, or v > 1, the equality holds if and only if P1(z) is1 + z1− z orone of its rotations.when 0 < v < 1, then the equality holds if and only ifP1(z) is1 + z21− z2 or one of its rotations. If v = 0, the equality holds if andonly if P1(z) =(12+λ2)1 + z1− z +(12− λ2)1− z1 + z. 0 ≤ λ ≤ 1 or one of itsrotations. If v = 1, the equality holds if and only if P1(z) is the reciprocalof one of the function for which the equality holds in the case of v = 0. Alsothe above upper bound can be improved as follows when 0 < v < 1|c2 − vc21|+ v|c1|2 ≤ 2 (0 < v ≤12),|c2 − vc21|+ (1− v)|c1|2 ≤ 2 (12< v ≤ 1).2. Series representation for universally prestarlike functionsTheorem 2.1. Let f be an universally prestarlike function of order0 ≤ α ≤ 1, then the function f(z) has a representation of the formf(z) = z +∞∑n=2anzn,wherean =n−1∑k=1C(α, k)akbn−kC′(α, n)− C(α, n), n = 2, 3, . . . (2.1)C(α, n) =n∏k=2(k − 2α)(n− 1)! , C(α, k) =k∏m=2(m− 2α)(k − 1)! , C(α, 1)a1 = 1FEKETE-SZEGO¨ INEQUALITY FOR . . . 389C′(α, n) =n∏k=2(k + 1− 2α)(n− 1)! , bn =∫ 10tndµ(t) and µ(t) is a probabilitymeasure on [0, 1].P r o o f. By Theorem 1.1, f ∈ Ruα if and only ifD3−2αfD2−2αf∈ T. Hence,D3−2αfD2−2αf=∫ 10dµ(t)1− tz , for some probability measure µ(t) on [0, 1],D3−2αfD2−2αf=∞∑n=0bnzn, where bn =∫ 10tndµ(t).Therefore,D3−2αf = z +∞∑n=2C′(α, n)anzn,where C′(α, n) =n∏k=2(k + 1− 2α)(n− 1)! , n = 2, 3, . . .Now,D2−2αf = z +∞∑n=2C(α, n)anzn,where C(α, n) =n∏k=2(k − 2α)(n− 1)! , n = 2, 3, . . .Therefore,D3−2αfD2−2αf=z +∞∑n=2C′(α, n)anznz +∞∑n=2C(α, n)anzn=∞∑n=0bnzn. (2.2)Equating the like of coefficients, we obtain for n = 2, 3, . . .:an =n−1∑k=1C(α, k)akbn−kC′(α, n)− C(α, n) ,with C(α, 1)a1 = 1.390 T.N. Shanmugam , J. Lourthu Mary3. Main resultWe now establish the Fekete Szego¨ inequality.Theorem 3.1. Let φ(z) = 1+B1z+B2z2+ . . . . If f(z) = z+∞∑n=2anznis a universally prestarlike function of order α, then|a3 − µa22| ≤B2 + (2− 2α)B21 − (3− 2α)B21µ3− 2α , µ ≤ σ1B13− 2α, σ1 ≤ µ ≤ σ2−B2 − (2− 2α)B21 + (3− 2α)B21µ3− 2α , µ ≥ σ2,where σ1 =(B2 −B1) + (2− 2α)B21(3− 2α)B21, σ2 =(B2 +B1) + (2− 2α)B21(3− 2α)B21.The result is sharp.P r o o f. If f ∈ Ruα, then there is a Schwartz function w(z), analytic in∆ with w(0) = 0 and |w(z)| < 1 in ∆ such that D3−2αfD2−2αf= φ(w(z)). Definethe function P1(z) by P1(z) =1 + w(z)1− w(z) = 1 + c1z + c2z2 + . . . . Since w(z)is a Schwartz function, we see that ReP1(z) > 0 and P1(0) = 1. Define thefunction P (z) =D3−2αfD2−2αf= 1+b1z+b2z2+. . . Now, P (z) = φ(P1(z)− 1P1(z) + 1),whereP1(z)− 1P1(z) + 1=c1z + c2z2 + . . .2 + c1z + c2z2 + . . .=12[c1z + z2[c2 − c212] + z3[c3 − c1c2 + c314] + . . .].Hence, on simplification, we getP (z) = 1 +B1c1z2+[B12(c2 − c212)+B2c214]z2 + . . .Therefore,1 + b1z + b2z2 + . . . = 1 +B1c1z2+[B12(c2 − c212)+B2c214]z2 + . . .Equating the like coefficients, we getFEKETE-SZEGO¨ INEQUALITY FOR . . . 391b1 =B1c12, (3.1)b2 =B12(c2 − c212)+B2c214. (3.2)Now,D3−2αfD2−2αf= 1 + b1z + b2z2 + . . . From equation (2.2), we have1 +[C′(α, 2)a2 − C(α, 2)a2] z + [C′(α, 3)a3 − C(α, 2)C′(α, 2)a22 − C(α, 3)a3+(C(α, 2)a2)2]z2 + . . . = 1 + b1z + b2z2 + . . .Equating the coefficients of z and z2 respectively and simplifying, we geta2 = b1 , a3 =b2 + (2− 2α)b213− 2α . (3.3)Applying equations (3.1) and (3.2) in (3.3), we geta2 =B1c12, a3 =13− 2α[B12(c2 − c212)+B2c214+ (2− 2α)B21c214].Now,a3 − µa22 =13− 2α[B12(c2 − c212)+B2c214+ (2− 2α)B21c214]− µB21c214=13− 2αB12[c2 − c21[12− B22B1− (2− 2α)B12+ (3− 2α)µB12]]=B12(3− 2α)[c2 − c21v],wherev =[12− B22B1− (2− 2α)B12+ (3− 2α)µB12].Now by an application of Lemma 1.1, if µ ≤ σ1,|a3 − µa22| ≤B2 + (2− 2α)B21 − (3− 2α)B21µ3− 2α ,whereσ1 =(B2 −B1) + (2− 2α)B21µ(3− 2α)B21.392 T.N. Shanmugam , J. Lourthu MaryNow, if σ1 ≤ µ ≤ σ2,|a3 − µa22| ≤B13− 2α.Now, if µ ≥ σ2,|a3 − µa22| ≤−B2 − (2− 2α)B21 + (3− 2α)B21µ3− 2α ,whereσ2 =(B2 +B1) + (2− 2α)B21µ(3− 2α)B21.If µ = σ1, then the equality holds in Lemma 1.1, if and only ifP1(z) =(12+λ2)1 + z1− z +(12− λ2)1− z1 + z, 0 ≤ λ ≤ 1,or one of its rotations. If µ = σ2, then1P1(z)=1(12+λ2)1 + z1− z +(12− λ2)1− z1 + z.If σ1 < µ < σ2, P1(z) =1 + λz21− λz2 . To show that the bounds are sharp, wedefine the function Kφnα (n = 2, 3, . . .) byD3−2αKφnαD3−2αKφnα= φ(zn−1),Kφnα (0) = 0, (Kφnα )′(0) = 1 and function F λα and Gλα (0 ≤ λ ≤ 1) by(D3−2αF λα)(z)(D2−2αF λα ) (z)= φ(z(z + λ)1 + λz),F λα (0) = 0, (Fλα )′(0) = 1 and similarly,(D3−2αGλα)(z)(D2−2αGλα) (z)= φ(z(z + λ)1 + λz)Gλα(0) = 0, (Gλα)′(0) = 1. Clearly, the functions Kφnα , F λα , Gλα ∈ Ruα. Also wewrite Kφα := Kφ2α . If µ < σ1 or µ < σ2, then the equality holds if and only iff is Kφα or one of its rotations. When σ1 < µ < σ2, then the equality holdsif and only if f is Kφ3α or one of its rotations. If µ = σ1, then the equalityholds if and only if f is F λα or one of its rotations If µ = σ2 then the equalityholds if and only if f is Gλα or one of its rotations. Hence the result follows.FEKETE-SZEGO¨ INEQUALITY FOR . . . 393Remark 3.1. If σ1 ≤ µ ≤ σ2, then in view of Lemma 1.1, Theorem 3.1can be improved. Let σ3 be given byσ3 =B2 + (2− 2α)B21(3− 2α)B21.If σ1 ≤ µ ≤ σ3, then|a3 − µa22|+((3− 2α)µB21 − [(B2 −B1) + (2− 2α)B21 ](3− 2α)B21)|a22| ≤B13− 2α.If σ2 ≤ µ ≤ σ3, then|a3 − µa22|+(−(3− 2α)µB21B2 +B1 + (2− 2α)B21(3− 2α)B21)|a22| ≤B13− 2α.P r o o f. For σ1 ≤ µ ≤ σ3, we have|a3 − µa22|+ (µ− σ1)|a22|=B12(3− 2α) |c2 − vc21|+(µ− [(B2 −B1) + (2− 2α)B21 ](3− 2α)B21)B21 |c1|24=B12(3− 2α)((3− 2α)µB21 −B2 −B1 − (2− 2α)B21(3− 2α)B21)B21 |c1|24=B1(3− 2α)[12|c2 − vc21|+12v|c1|2]=B1(3− 2α)[12[|c2 − vc21|+ v|c1|2]].Now, by using Lemma 1.1, we get |a3−µa22|+(µ−σ1)|a22| ≤B1(3− 2α) . Now,for σ2 ≤ µ ≤ σ3 , we have|a3 − µa22|+ (σ2 − µ)|a22|=B12(3− 2α) |c2 − vc21|+(B2 +B1 + (2− 2α)B21(3− 2α)B21− µ)B21 |c1|24=B12(3− 2α)(−(3− 2α)µB21 +B2 +B1 + (2− 2α)B21(3− 2α)B21)B21 |c1|24=B1(3− 2α)(12[|c2 − vc21|+ (1− v)|c1|2]).394 T.N. Shanmugam , J. Lourthu MaryNow, by using Lemma 1.1, we get |a3 − µa22|+ (σ2 − µ)|a22| ≤B1(3− 2α) .Hence the result follows.References[1] W. Ma, D. Minda, A unified treatment of some special classes of univalentfunctions. Proc. Conf. on Complex Analysis (Eds: F. Li, L. Ren, L.Yang, S. Zhang), Int. Press (1994), 157-169.[2] S. Ruscheweyh, Some properties of prestarlike and universally prestarlikefunctions. Intern. Conf. Geometric Function Theory, J. Analysis 15(2007), 247-254.[3] S. Ruscheweyh, L. Salinas, Universally prestarlike functions as convolu-tion multipliers. Mathematische Zeitschrift (2009), 607-617.[4] S. Ruscheweyh, L. Salinas, T. Sugawa, Completely monotone sequencesand universally prestarlike functions. Israel J. Math. 171 (2009), 285-304.Department of MathematicsAnna University Chennai, Chennai – 600025, INDIA∗ e-mail: shan@annauniv.edu∗∗ e-mail: j lourthumary@annauniv.eduReceived: GFTA, August 27-31, 2010

Fekete-Szegö Inequality for Universally Prestarlike Functions

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Fekete-Szegö Inequality for Universally Prestarlike Functions

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