In this paper, we introduce a new general integral operator
and obtain the order of convexity of this integral operator

Bulut, Serap

Okayama University Digital Information Repository

CONVEXITY PROPERTIES OF A NEW GENERAL INTEGRAL OPERATOR OF p-VALENT FUNCTIONS

Okayama University Scientific Achievement Repository

Math. J. Okayama Univ. 56 (2014), 171–178CONVEXITY PROPERTIES OF A NEW GENERALINTEGRAL OPERATOR OF p-VALENT FUNCTIONSSerap BULUTAbstract. In this paper, we introduce a new general integral operatorand obtain the order of convexity of this integral operator.1. Introduction and definitionsLet Ap denote the class of all functions of the form(1.1) f(z) = zp +∞∑k=p+1akzk (p ∈ N = {1, 2, . . .})which are analytic in the open unit disk U := {z ∈ C : |z| < 1}. In particular,we set A1 ≡ A.A function f ∈ Ap is said to be p-valently starlike of order γ (0 ≤ γ < p)if and only if f satisfies(1.2) ℜ{zf ′(z)f(z)}> γfor all z ∈ U. We say that f is in the class S∗p(γ) for such functions. Inparticular, we set S∗p(0) ≡ S∗p for p-valently starlike functions in U.On the other hand, a function f ∈ Ap is said to be p-valently convex oforder γ (0 ≤ γ < p) if and only if f satisfies(1.3) ℜ{1 +zf ′′(z)f ′(z)}> γfor all z ∈ U. We say that f is in the class Kp(γ) for such functions. Inparticular, we set Kp(0) ≡ Kp for p-valently convex functions in U.Also, we note that S∗1 (γ) ≡ S∗(γ) and K1(γ) ≡ K(γ) are the classes ofstarlike and convex functions of order γ (0 ≤ γ < 1), respectively.A function f ∈ Ap is in the class Rp(γ) if it satisfiesℜ{f ′(z)zp−1}> γfor all z ∈ U.Mathematics Subject Classification. Primary 30C45.Key words and phrases. Analytic function, Multivalent function, Starlikefunction, Convex function, Integral operator.171172 S. BULUTA function f ∈ Ap is in the class USp(δ, γ) if and only if f satisfies(1.4) ℜ{zf ′(z)f(z)}> δ∣∣∣∣zf′(z)f(z)− p∣∣∣∣+ γ,where δ ≥ 0, γ ∈ [−1, p), δ + γ ≥ 0, z ∈ U.Furthermore, a function f ∈ Ap is in the class UKp(δ, γ) if and only if fsatisfies(1.5) ℜ{1 +zf ′′(z)f ′(z)}> δ∣∣∣∣zf′′(z)f ′(z)− (p− 1)∣∣∣∣+ γ,where δ ≥ 0, γ ∈ [−1, p), δ + γ ≥ 0, z ∈ U.Now we consider following comprehensive class:A function f ∈ Ap is in the class Bp(µ, γ) if and only if f satisfies∣∣∣∣f′ (z)zp−1(zpf (z))µ− p∣∣∣∣ < p− γ,where µ ≥ 0, γ ∈ [0, p), z ∈ U.Remark 1. This family is a comprehensive class of analytic functionsthat contains other new classes of analytic functions as well as some verywell-known ones. For example,(i) For µ = 1, we have the classBp(1, γ) ≡ S∗p(γ).(ii) For µ = 0, we have the classBp(0, γ) ≡ Rp(γ).(iii) For p = 1, this class studied by Frasin and Jahangiri [11].(iv) For p = 1 and µ = 2, this class studied by Frasin and Darus [10].Let αi, βi ∈ C and fi, gi ∈ Ap for all i = 1, 2, . . . , n, n ∈ N. We define thefollowing general integral operator(1.6) Iα,βn (f, g) (z) =∫ z0ptp−1n∏i=1(fi (t)tp)αi ( g′i (t)ptp−1)βidt.Remark 2. (i) For p = 1, we have the integral operatorK (z) =∫ z0n∏i=1(fi (t)t)αi (g′i (t))βi dtintroduced and studied by Ularu [13].CONVEXITY PROPERTIES OF A NEW GENERAL INTEGRAL OPERATOR 173(ii) For αi > 0, βi = 0 (1 ≤ i ≤ n) and αi = 0, βi > 0 (1 ≤ i ≤ n), wehave the integral operatorsFp (z) =∫ z0ptp−1(f1 (t)tp)α1· · ·(fn (t)tp)αndtandGp (z) =∫ z0ptp−1(g′1 (t)ptp−1)β1· · ·(g′n (t)ptp−1)βndt,respectively. This integral operators are introduced and studied by Frasin[8, 9]. Also for p = 1, the integral operatorsFn (z) =∫ z0(f1 (t)t)α1· · ·(fn (t)t)αndtandFβ1,...,βn (z) =∫ z0(g′1 (t))β1 · · · (g′n (t))βn dtstudied recently by many authors (see [1]-[7]).The following result will be required in our investigation.General Schwarz Lemma. [12] Let the function f be regular in thedisk UR = {z ∈ C : |z| < R}, with |f(z)| < M for fixed M . If f has onezero with multiplicity order bigger than m for z = 0, then|f(z)| ≤MRm|z|m (z ∈ UR).The equality can hold only if f(z) = eiθ(M/Rm) zm, where θ is constant.2. Main ResultsTheorem 2.1. Let αi, βi ≥ 0, δi ≥ 0, γi ∈ [−1, p), δi + γi ≥ 0 and fi ∈USp(δi, γi), gi ∈ UKp(δi, γi) for all i = 1, 2, . . . , n. If(2.1) 0 ≤ p+n∑i=1(αi + βi) (γi − p) ≤ p,then the integral operator Iα,βn (f, g) defined by (1.6) is p-valently convex oforder λ withλ = p+n∑i=1(αi + βi) (γi − p) .174 S. BULUTProof. From (1.6), it is easy to see that(2.2)(Iα,βn (f, g))′(z) = pzp−1n∏i=1(fi (z)zp)αi ( g′i (z)pzp−1)βi.Differentiating both sides of (2.2) logarithmically and after some calculus,we obtain1 +z(Iα,βn (f, g))′′(z)(Iα,βn (f, g))′(z)(2.3)= p+n∑i=1αi(zf ′i (z)fi (z))+n∑i=1βi(1 +zg′′i (z)g′i (z))−n∑i=1p (αi + βi) .Since fi ∈ USp(δi, γi) and gi ∈ UKp(δi, γi) for all i = 1, 2, . . . , n, it followsfrom (1.4) and (1.5) thatℜ1 +z(Iα,βn (f, g))′′(z)(Iα,βn (f, g))′(z)(2.4)= p+n∑i=1αiℜ{zf ′i (z)fi (z)}+n∑i=1βiℜ{1 +zg′′i (z)g′i (z)}−n∑i=1p (αi + βi)>n∑i=1αiδi∣∣∣∣zf′i (z)fi (z)− p∣∣∣∣+n∑i=1βiδi∣∣∣∣zg′′i (z)g′i (z)− (p− 1)∣∣∣∣+ p+n∑i=1(αi + βi) (γi − p) .Becauseαiδi∣∣∣∣zf′i (z)fi (z)− p∣∣∣∣ ≥ 0andβiδi∣∣∣∣zg′′i (z)g′i (z)− (p− 1)∣∣∣∣ ≥ 0for all i = 1, 2, . . . , n, from (2.4), we obtainℜ1 +z(Iα,βn (f, g))′′(z)(Iα,βn (f, g))′(z) > p+n∑i=1(αi + βi) (γi − p) .CONVEXITY PROPERTIES OF A NEW GENERAL INTEGRAL OPERATOR 175Therefore Iα,βn (f, g) is p-valently convex of orderλ = p+n∑i=1(αi + βi) (γi − p) .Remark 3. (i) Letting p = 1 in Theorem 2.1, we obtain Theorem 2.2 in[13].(ii) Letting β1 = · · · = βn = 0 and α1 = · · · = αn = 0 in Theorem 2.1, weobtain Theorem 2.1 and Theorem 3.1 in [9], respectively.Letting n = 1, p = 1, α1 = α, β1 = β, δ1 = δ, γ1 = γ and f1 = f , g1 = gin Theorem 2.1, we haveCorollary 2.2. Let α, β ≥ 0, δ ≥ 0, γ ∈ [−1, 1), δ + γ ≥ 0 and f ∈USp(δ, γ), g ∈ UKp(δ, γ). If0 ≤ 1 + (α+ β) (γ − 1) ≤ 1,then the integral operator(2.5) Iα,β (f, g) (z) =∫ z0(f (t)t)α (g′ (t))βdtis convex of order1 + (α+ β) (γ − 1) .Theorem 2.3. Let αi, βi ≥ 0, δi ≥ 0, γi ∈ [−1, p), δi + γi ≥ 0 and(2.6)∣∣∣∣zf′i (z)fi (z)− p∣∣∣∣ > −p2+∑ni=1 αi (γi − p)∑ni=1 αiδi,(2.7)∣∣∣∣zg′′i (z)g′i (z)− (p− 1)∣∣∣∣ > −p2+∑ni=1 βi (γi − p)∑ni=1 βiδifor all i = 1, 2, . . . , n, then the integral operator Iα,βn (f, g) defined by (1.6)is p-valently convex in U.Proof. From (2.4), (2.6) and (2.7), we easily get Iα,βn (f, g) ∈ Kp. Theorem 2.4. Let fi, gi ∈ A, where fi ∈ B(µi, γi) (µi ≥ 0, 0 ≤ γi < p),αi, βi ∈ C and Mi ≥ 1 for all i = 1, . . . , n. If|fi (z)| < Mi,∣∣∣∣g′′i (z)g′i (z)∣∣∣∣ ≤ 1176 S. BULUTfor all i = 1, . . . , n, and if0 ≤ p−n∑i=1|αi|(p+ (2p− γi)Mµi−1i)−n∑i=1|βi| (p+ 2) < p,then the integral operator Iα,βn (f, g) defined by (1.6) is p-valently convex oforder η withη = p−n∑i=1|αi|(p+ (2p − γi)Mµi−1i)−n∑i=1|βi| (p+ 2) .Proof. From (2.3) we have1 +z(Iα,βn (f, g))′′(z)(Iα,βn (f, g))′(z)− p=n∑i=1αi(zf ′i (z)fi (z))+n∑i=1βi(1 +zg′′i (z)g′i (z))−n∑i=1p (αi + βi) .This implies that∣∣∣∣∣∣∣1 +z(Iα,βn (f, g))′′(z)(Iα,βn (f, g))′(z)− p∣∣∣∣∣∣∣(2.8)≤n∑i=1|αi|∣∣∣∣zf′i (z)fi (z)∣∣∣∣+n∑i=1|βi| |z|∣∣∣∣g′′i (z)g′i (z)∣∣∣∣+n∑i=1p |αi|+n∑i=1|βi| (p+ 1)=n∑i=1|αi|∣∣∣∣f′i (z)zp−1(zpfi (z))µi∣∣∣∣∣∣∣∣fi (z)zp∣∣∣∣µi−1+n∑i=1|βi| |z|∣∣∣∣g′′i (z)g′i (z)∣∣∣∣+n∑i=1p |αi|+n∑i=1|βi| (p+ 1) .Sincefi ∈ B(µi, γi),|fi (z)| < Miand ∣∣∣∣g′′i (z)g′i (z)∣∣∣∣ ≤ 1CONVEXITY PROPERTIES OF A NEW GENERAL INTEGRAL OPERATOR 177for all i = 1, . . . , n, applying General Schwarz Lemma and using (2.8), weobtain ∣∣∣∣∣∣∣1 +z(Iα,βn (f, g))′′(z)(Iα,βn (f, g))′(z)− p∣∣∣∣∣∣∣≤n∑i=1|αi|(∣∣∣∣f′i (z)zp−1(zpfi (z))µi− p∣∣∣∣Mµi−1i + pMµi−1i)+n∑i=1|βi| |z|+n∑i=1p |αi|+n∑i=1|βi| (p+ 1)<n∑i=1|αi|(p+ (2p − γi)Mµi−1i)+n∑i=1|βi| (p+ 2) = p− η.This implies that the integral operator Iα,βn (f, g) is p-valently convex oforderη = p−n∑i=1|αi|(p+ (2p − γi)Mµi−1i)−n∑i=1|βi| (p+ 2) .Letting n = 1, p = 1, α1 = α, β1 = β, µ1 = µ, γ1 = γ, M1 = M andf1 = f , g1 = g in Theorem 2.4, we haveCorollary 2.5. Let f, g ∈ A, where f ∈ B(µ, γ) (µ ≥ 0, 0 ≤ γ < p),α, β ∈ C and M ≥ 1. If|f (z)| < M,∣∣∣∣g′′ (z)g′ (z)∣∣∣∣ ≤ 1and if0 ≤ 1− |α|(1 + (2− γ)Mµ−1)− 3 |β| < 1,then the integral operator Iα,β (f, g) defined by (2.5) is convex of order1− |α|(1 + (2− γ)Mµ−1)− 3 |β| .References[1] D. Breaz, A convexity property for an integral operator on the class Sp (β), J. Inequal.Appl.(2008), Art. ID 143869.[2] D. Breaz, Certain integral operators on the classes M (βi) and N (βi), J. Inequal.Appl. (2008), Art. ID 719354.[3] D. Breaz and N. Breaz, Two integral operators, Studia Univ. Babes-Bolyai Math. 47(2002), no. 3, 13-19.178 S. BULUT[4] D. Breaz and H.O¨. Gu¨ney, The integral operator on the classes S∗α (b) and Cα (b), J.Math. Ineq. 2 (2008), no. 1, 97-100.[5] D. Breaz, S. Owa and N. Breaz, A new integral univalent operator, Acta Univ. Apu-lensis Math. Inform. 16 (2008), 11-16.[6] S. Bulut, A note on the paper of Breaz and Gu¨ney, J. Math. Ineq. 2 (2008), no. 4,549-553.[7] B.A. Frasin, Some sufficient conditions for certain integral operators, J. Math. Ineq. 2(2008), no. 4, 527-535.[8] B.A. Frasin, New general integral operators of p-valent functions, J. Inequal. PureAppl. Math. 10 (2009), no. 4, Art. 109, 9 pp.[9] B.A. Frasin, Convexity of integral operators of p-valent functions, Math. Comput.Modelling 51 (2010), no. 5-6, 601-605.[10] B.A. Frasin and M. Darus, On certain analytic univalent functions, Int. J. Math.Math. Sci. 25 (2001), no. 5, 305-310.[11] B.A. Frasin and Jay M. Jahangiri, A new and comprehensive class of analytic func-tions, An. Univ. Oradea Fasc. Mat. 15 (2008), 59-62.[12] Z. Nehari, Conformal Mapping, Dover, New York, NY, USA, 1975.[13] N. Ularu, Convexity properties for an integral operator, Acta Univ. Apulensis Math.Inform. No. 27 (2011), 115-120.Serap BULUTKocaeli UniversityCivil Aviation CollegeArslanbey CampusI˙zmit-Kocaeli, 41285 Turkeye-mail address: serap.bulut@kocaeli.edu.tr(Received December 6, 2011 )

CONVEXITY PROPERTIES OF A NEW GENERAL INTEGRAL OPERATOR OF p-VALENT FUNCTIONS

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