Here we investigate a majorization problem involving starlike meromorphic functions of complex order belonging to a certain subclass of meromorphic univalent functions defined by an integral operator introduced recently by Lashin

Goswami, Pranay

Goyal, S. P.

English

University of Maria Curie-Skłodowska (UMCS): Scientific e-Journals / Uniwersytet Marii Curie-Skłodowskiej: e-czasopisma naukowe

doi: 10.2478/v10062-012-0013-1ANNALESU N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K AL U B L I N – P O L O N I AVOL. LXVI, NO. 2, 2012 SECTIO A 57–62S. P. GOYAL and PRANAY GOSWAMIMajorization for certain classesof meromorphic functions definedby integral operatorAbstract. Here we investigate a majorization problem involving starlikemeromorphic functions of complex order belonging to a certain subclass ofmeromorphic univalent functions defined by an integral operator introducedrecently by Lashin.1. Introduction and preliminaries. Let f(z) and g(z) be analytic inthe open unit disk(1.1) 4 = {z ∈ C and |z| < 1}.For analytic functions f(z) and g(z) in ∆, we say that f(z) is majorizedby g(z) in ∆ (see [9]) and write(1.2) f(z) g(z) (z ∈ ∆),if there exists a function φ(z), analytic in ∆ such that |φ(z)| ≤ 1, and(1.3) f(z) = φ(z)g(z) (z ∈ ∆).Let Σ denote the class of meromorphic functions of the form(1.4) f(z) =1z+∞∑k=1akzk,2000 Mathematics Subject Classification. Primary 30C45; Secondary 30C80.Key words and phrases. Meromorphic univalent functions, majorization property, star-like functions, integral operators.58 S. P. Goyal and P. Goswamiwhich are analytic and univalent in the punctured unit disk(1.5) ∆∗ := {z ∈ C : 0 < |z| < 1} := ∆ \ {0}with a simple pole at the origin.For functions fj ∈ Σ given by(1.6) fj(z) =1z+∞∑k=1ak,jzk (j = 1, 2; z ∈ ∆∗),we define the Hadamard product (or convolution) of f1 and f2 by(1.7) (f1 ∗ f2)(z) = 1z+∞∑k=1ak,1ak,2zk = (f2 ∗ f1)(z).Analogously to the operators defined by Jung, Kim and Srivastava [7] on thenormalized analytic functions, Lashin [8] introduced the following integraloperatorsPαβ : Σ −→ Σdefined by(1.8) Pαβ = Pαβ f(z) =βαΓ(α)1zβ+1z∫0tβ(logzt)α−1f(t)dt(α > 0, β > 0; z ∈ ∆∗), where Γ(α) is the familiar Gamma function.Using the integral representation of the Gamma function and (1.4), it canbe easily shown that(1.9) Pαβ f(z) =1z+∞∑k=1(ββ + k + 1)αakzk, (α > 0, β > 0; z ∈ ∆∗).Obviously(1.10) P1βf(z) := Jβ.The operatorJβ : Σ −→ Σhas also been studied by Lashin [8].It is easy to verify that (see [8]),(1.11) z(Pαβ f(z))′ = βPα−1β f(z)− (β + 1)Pαβ f(z).Definition 1.1. A function f(z) ∈ Σ is said to be in the class Sα,jβ (γ) ofmeromorphic functions of complex order γ 6= 0 in ∆ if and only if(1.12) <{1− 1γ(z(Pαβ f(z))(j+1)(Pαβ f(z))(j)+ j + 1)}> 0(z ∈ ∆, j ∈ N0 = N ∪ {0}, α > 0, β > 0, γ ∈ C\{0}).Majorization for certain classes of meromorphic functions... 59Clearly, we have the following relationships:(i) S0,0β (γ) = S(γ) (γ ∈ C \ {0}),(ii) S0,0β (1− η) = S∗(η) (0 ≤ η < 1).The classes S(γ) and S∗(η) are said to be classes of meromorphic star-like univalent functions of complex order γ 6= 0 and meromorphic starlikeunivalent functions of order η (η ∈ < such that 0 ≤ η < 1) in ∆∗.A majorization problem for the normalized classes of starlike functionshas been investigated by Altinas et al. [1] and MacGregor [9]. In the re-cent paper Goyal and Goswami [2] generalized these results for the class ofmultivalent functions, using fractional derivatives operators. Further, Goyalet al. [3], Goswami and Wang [4], Goswami [5], Goswami et al. [6] studiedmajorization property for different classes. In this paper, we will study ma-jorization properties for the class of meromorphic functions using integraloperator Pαβ .2. Majorization problems for the class Sα,jβ (γ).Theorem 2.1. Let the function f ∈ Σ and suppose that g ∈ Sα,jβ (γ). If(Pαβ f(z))(j) is majorized by (Pαβ g(z))(j) in ∆∗, then(2.1) |(Pα−1β f(z))(j)| ≤ |(Pα−1β g(z))(j)| for |z| ≤ r1(β, γ),where(2.2) r1(β, γ) =k1 −√k12 − 4β|β + 2γ|2|β + 2γ|andk1 = β + 2 + |β + 2γ|, (β > 0, j ∈ N0, γ ∈ C \ {0}).Proof. Since g ∈ Sα,jβ (γ), we find from (2.1) that if(2.3) h1(z) = 1− 1γ(z(Pαβ g(z))(j+1)(Pαβ g(z))(j)+ j + 1)(α, β > 0, γ ∈ C \ {0}, j ∈ N0), then <{h1(z)} > 0 (z ∈ ∆) and(2.4) h1(z) =1 + w(z)1− w(z) (w ∈ P),where P denotes the well-known class of bounded analytic functions in ∆and w(z) = c1z + c2z2 + . . . satisfies the conditionsw(0) = 0 and |w(z)| ≤ |z| (z ∈ ∆).60 S. P. Goyal and P. GoswamiMaking use of (2.3) and (2.4), we get(2.5)z(Pαβ g(z))(j+1)(Pαβ g(z))(j)=(1 + j − 2γ)w(z)− (1 + j)1− w(z) .By the principle of mathematical induction, and (1.11), we easily get(2.6) z(Pαβ g(z))(j+1) = β(Pα−1β g(z))(j) − (β + j + 1)(Pαβ g(z))(j),(α > 1, β > 0; z ∈ ∆∗). Now using (2.6) in (2.5), we find thatβ(Pα−1β g(z))(j)(Pαβ g(z))(j)= (β + j + 1) +(1 + j − 2γ)w(z)− (1 + j)1− w(z)=β − (β + 2γ)w(z)1− w(z)or(2.7) (Pαβ g(z))(j) =β(1− w(z))β − (β + 2γ)w(z)(Pα−1β g(z))(j).Since |w(z)| ≤ |z| (z ∈ ∆), the formula (2.6) yields(2.8)∣∣∣(Pαβ g(z))(j)∣∣∣ ≤ β[1 + |z|]β − |β + 2γ||z| ∣∣∣(Pα−1β g(z))(j)∣∣∣ .Next since (Pαβ f(z))(j) is majorized by (Pαβ g(z))(j) in the unit disk ∆∗, from(1.3), we have(Pαβ f(z))(j) = φ(z)(Pαβ g(z))(j).Differentiating it with respect to z and multiplying by z, we getz(Pαβ f(z))(j+1) = zϕ′(z)(Pαβ g(z))(j) + zϕ(z)(Pαβ g(z))(j+1).Using (2.7), in the above equality, it yields(2.9) (Pα−1β f(z))(j) =zϕ′(z)β(Pαβ g(z))(j) + ϕ(z)(Pα−1β g(z))(j).Thus, nothing that ϕ ∈ P satisfies the inequality (see, e.g. Nehari [6])(2.10)∣∣ϕ′(z)∣∣ ≤ 1− |ϕ(z)|21− |z|2and making use of (2.8) and (2.10) in (2.9), we get(2.11)∣∣∣(Pα−1β f(z))(j)∣∣∣≤(|ϕ(z)|+ 1− |ϕ(z)|21− |z||z|[β − |2γ + β||z|]) ∣∣∣(Pα−1β g(z))(j)∣∣∣ ,which upon setting|z| = r and |ϕ(z)| = ρ (0 ≤ ρ ≤ 1),Majorization for certain classes of meromorphic functions... 61leads us to the inequality∣∣∣((Pα−1β f(z))(j))∣∣∣ ≤ Θ(ρ)(1− r)(β − |2γ + β|r) ∣∣∣(Pα−1β g(z))(j)∣∣∣ ,where(2.12) Θ(ρ) = −rρ2 + (1− r)(β − |2γ + β|r)ρ+ rtakes its maximum value at ρ = 1, with r2 = r2(β, γ), where r2(β, γ) isgiven by equation (2.2). Furthermore, if 0 ≤ ρ ≤ r2(β, γ), then the functionθ(ρ) defined by(2.13) θ(ρ) = −σρ2 + (1− σ)(β − |2γ + β|σ)ρ+ σis an increasing function on the interval 0 ≤ ρ ≤ 1, so that(2.14) θ(ρ) ≤ θ(1) = (1− σ)(β − |2γ + β|σ),(0 ≤ ρ ≤ 1; 0 ≤ σ ≤ r1(β, γ)). Hence upon setting ρ = 1 in (2.14), weconclude that (2.1) of Theorem 2.1 holds true for |z| ≤ r1(β, γ), wherer1(β, γ) is given by (2.2). This completes the proof of Theorem 2.1. Setting α = 1 in Theorem 2.1, we getCorollary 2.1. Let the function f ∈ Σ and suppose that g ∈ S1,jβ (γ). If(Jβf(z))(j) is majorized by (Jβg(z))(j) in ∆∗, then(2.15) |(f(z))(j)| ≤ |(g(z))(j)| for |z| ≤ r2(β, γ),wherer2(β, γ) =k2 −√k22 − 4β|β + 2γ|2|β + 2γ|andk2 = β + 2 + |β + 2γ|, (β > 0, j ∈ N0, γ ∈ C\{0}).Further putting β = 1 and γ = 1− η, j = 0 in Corollary 2.1, we getCorollary 2.2. Let the function f ∈ Σ and suppose that g ∈ S1,01 (1 − η).If (J1f(z)) is majorized by (J1g(z)) in ∆∗, then(2.16) |f(z)| ≤ |g(z)| for |z| ≤ r3,wherer3 =3− η −√η2 − 4η + 63− η .For η = 0, the above corollary reduces to the following result:Corollary 2.3. Let the function f(z) ∈ Σ and suppose that g ∈ S1,01 (1) :=S1,01 . If (J1f(z)) is majorized by (J1g(z)) in ∆∗, then(2.17) |f(z)| ≤ |g(z)| for |z| ≤ 3−√63.62 S. P. Goyal and P. GoswamiAcknowledgments. The first author is thankful to CSIR, New Delhi, In-dia for awarding Emeritus Scientist under scheme No. 21(084)/10/EMR-II.References[1] Altintas¸, O., O¨zkan, O¨., Srivastava, H. M., Majorization by starlike functions ofcomplex order, Complex Variables Theory Appl. 46 (2001), 207–218.[2] Goyal, S. P., Goswami, P., Majorization for certain classes of analytic functionsdefined by fractional derivatives, Appl. Math. Lett. 22 (12) (2009), 1855–1858.[3] Goyal, S. P., Bansal S. K., Goswami, P., Majorization for certain classes of analyticfunctions defined by linear operator using differential subordination, J. Appl. Math.Stat. Inform. 6 (2) (2010), 45–50.[4] Goswami, P., Wang, Z.-G., Majorization for certain classes of analytic functions,Acta Univ. Apulensis Math. Inform. 21 (2009), 97–104.[5] Goswami, P., Aouf, M. K., Majorization properties for certain classes of analyticfunctions using the Sa˘la˘gean operator, Appl. Math. Lett. 23 (11) (2010), 1351–1354.[6] Goswami, P., Sharma, B., Bulboaca˘, T., Majorization for certain classes of analyticfunctions using multiplier transformation, Appl. Math. Lett. 23 (10) (2010), 633–637.[7] Jung, I. B., Kim, Y. C., Srivastava, H. M., The Hardy space of analytic functionsassociated with certain one-parameter families of integral operator, J. Math. Anal.Appl. 176 (1) (1993), 138–147.[8] Lashin, A. Y., On certain subclasses of meromorphic functions associated with certainintegral operators, Comput. Math. Appl., 59 (1) (2010), 524–531.[9] MacGreogor, T. H., Majorization by univalent functions, Duke Math. J. 34 (1967),95–102.[10] Nehari, Z., Conformal Mapping, MacGraw-Hill Book Company, New York, Torontoand London, 1955.S. P. Goyal Pranay GoswamiDepartment of Mathematics Department of MathematicsUniversity of Rajasthan AMITY University RajasthanJaipur-302055 Jaipur-302002India Indiae-mail: somprg@gmail.com e-mail: pranaygoswami83@gmail.comReceived March 7, 2011

Majorization for certain classes of meromorphic functions defined by integral operator

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Majorization for certain classes of meromorphic functions defined by integral operator

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