In this paper we discuss some subordination results for a subclass of functions analytic in the unit disk U

Opoola, T. O.

Oyekan, E. A.

English

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

doi: 10.2478/umcsmath-2014-0007ANNALESUNIVERS ITAT I S MARIAE CUR IE - SKŁODOWSKALUBL IN – POLONIAVOL. LXVIII, NO. 1, 2014 SECTIO A 59–66E. A. OYEKAN and T. O. OPOOLAOn a subordination result for analytic functionsdefined by convolutionAbstract. In this paper we discuss some subordination results for a subclassof functions analytic in the unit disk U .1. Introduction. Let A be the class of functions f(z) analytic in the unitdisk U = {z : |z| < 1} and normalized by(1.1) f(z) = z +∞∑n=2anzn.We denote by K(α) the class of convex functions of order α, i.e.,K(α) ={f ∈ A : Re(1 +zf′′(z)f ′(z))> α, z ∈ U}.Definition 1 (Hadamard product or convolution). Given two functionsf(z) and g(z), where f(z) is defined in (1.1) and g(z) is given byg(z) = z +∞∑n=2bnzn,2000 Mathematics Subject Classification. 30C45, 30C80.Key words and phrases. Subordination, analytic functions, Hadamard product(convolution).60 E. A. Oyekan and T. O. Opoolathe Hadamard product (or convolution) f ∗ g of f(z) and g(z) is defined by(1.2) (f ∗ g)(z) = z +∞∑n=2anbnzn = (g ∗ f)(z).Definition 2 (Subordination). Let f(z) and g(z) be analytic in the unitdisk U. Then f(z) is said to be subordinate to g(z) in U and we writef(z) ≺ g(z), z ∈ U,if there exists a Schwarz function w(z), analytic in U with w(0) = 0, |w(z)| <1 such that(1.3) f(z) = g(w(z)), z ∈ U.In particular, if the function g(z) is univalent in U , then f(z) is subordinateto g(z) if(1.4) f(0) = g(0), f(U) ⊆ g(U).Definition 3 (Subordinating factor sequence). A sequence {bn}∞n=1 of com-plex numbers is said to be a subordinating factor sequence if whenever f(z)of the form (1.1) is analytic, univalent and convex in U , the subordinationis given by∞∑n=1anbnzn ≺ f(z), z ∈ U, a1 = 1.We have the following theorem.Theorem 1.1 (Wilf [5]). The sequence {bk}∞k=1 is a subordinating factorsequence if and only if(1.5) Re{1 + 2∞∑k=1bkzk}> 0, z ∈ U.Let(1.6) M(α) ={f ∈ A : Re(zf ′(z)f(z))< α, z ∈ U}and let(1.7) M δ(b, δ) ={f ∈ A : Re{1− 25+2Dδ+2f(z)bDδ+1f(z)}< α, α > 0, z ∈ U}.On a subordination result for analytic functions... 61Here Dδf(z) is the Ruschewey’s derivative defined asDδf(z) =z(1− z)δ+1 ∗ f(z)=(z +∞∑n=2Γ(n+ δ)(n− 1)!Γ(1 + δ))∗(z +∞∑n=2anzn)= z +∞∑n=2Γ(n+ δ)(n− 1)!Γ(1 + δ)anzn, δ ≥ −1.Theorem 1.2 ([3]). If f(z) ∈ A satisfies∞∑n=2{|b(1− k)(δ + 2) + 2(n− 1)|+ |b(1− 2α+ k)(δ + 2)+ 2(n− 1)|} Γ(n+ δ + 1)(n− 1)!Γ(3 + δ) |an| ≤ 2|b(1− α)|(1.8)where b is a non-zero complex number, δ ≥ −1, 0 ≤ k ≤ 1 and α > 1, thenf(z) ∈ M δ(b, α).It is natural to consider the class M δ∗(b, α) ⊂ M δ(b, α) such thatM δ∗(b, α) ={f ∈ A :∞∑n=2{|b(1− k)(δ + 2) + 2(n− 1)|+ |b(1− 2α+ k)(δ + 2) + 2(n− 1)|} Γ(n+ δ + 1)(n− 1)!Γ(3 + δ) |an|≤ |b(1− α)|}.(1.9)Our main result in this paper is the following theorem.Theorem 1.3. Let f ∈ M δ∗(b, α), then(1.10)BC(f ∗ g)(z) ≺ g(z)whereB = |b(1− k)(δ + 2) + 2|+ |b(1− 2α+ k)(δ + 2) + 2|C = 2[2|b(1− α)|+ |b(1− k)(δ + 2) + 2|+ |b(1− 2α+ k)(δ + 2) + 2|],δ ≥ −1, 0 ≤ k ≤ 1, b is a non-zero complex number and g(z) ∈ K(α),z ∈ U . Moreover,(1.11) Re(f(z)) > − C2B.The constant factorBC=|b(1− k)(δ + 2) + 2|+ |b(1− 2α+ k)(δ + 2) + 2|2[2|b(1− α)|+ |b(1− k)(δ + 2) + 2|+ |b(1− 2α+ k)(δ + 2) + 2|]62 E. A. Oyekan and T. O. Opoolacannot be replaced by a larger one.2. Proof of the main result. Let f(z) ∈ M δ∗(b, α) and suppose thatg(z) = z +∞∑n=2bnzn ∈ K(α).Then by definition,(2.1)BC(f ∗ g)(z) = BC(z +∞∑n=2anbnzn).Hence, by Definition 3, to show the subordination (1.10) it is enough toprove that(2.2){BCan}∞n=1is a subordinating factor sequence with a1 = 1. Therefore, by Theorem 1.1it is sufficient to show that(2.3) Re{1 + 2∞∑n=1BCanzn}> 0, z ∈ U.Now,Re{1 + 2∞∑n=1BCanzn}= Re{1 + 2BCa1z +2C∞∑n=2Banzn}≥ 1− 2BCr − 2C∞∑n=2B|an|rn.(2.4)SinceΓ(n+ δ + 1)(n− 1)!Γ(3 + δ) is a monotone non-decreasing function of n= 2, 3, . . . ,we haveRe{1 + 2∞∑n=1BCanzn}> 1− 2BCr− 2C∞∑n=2{[|b(1− k)(δ + 2) + 2(n− 1)]|+ |b(1− 2α+ k)(δ + 2) + 2(n− 1)|}× Γ(n+ δ + 1)(n− 1)!Γ(3 + δ) |an|r, 0 < r < 1.By (1.8)∞∑n=2{|b(1− k)(δ + 2) + 2(n− 1)|+ |b(1− 2α+ k)(δ + 2) + 2(n− 1)|}× Γ(n+ δ + 1)(n− 1)!Γ(3 + δ) |an| ≤ 2|b(1− α)|.On a subordination result for analytic functions... 63Hence,Re{1 + 2∞∑n=1BCanzn}= Re{1 + 2BCa1z +2C∞∑n=2Banzn}> 1− 2BCr − 4|b(1− α)|Cr= 1− 2B + 4|b(α− 1)|Cr= 1− r > 0(|z| = r < 1). Therefore, we obtainRe{1 + 2∞∑n=1BCanzn}> 0which is (2.3) that was to be established.We now show thatRe(f(z)) > − C2B.Takingg(z) =z1− z ∈ K(α),(1.10) becomesBCf(z) ≺ z1− z .Therefore,(2.5) Re(BCf(z))> Re(z1− z).Since(2.6) Re(z1− z)> −12, |z| < r,this implies that(2.7)BCRe(f(z)) > −12.Hence, we haveRe(f(z)) > − C2Bwhich is (1.11).To show the sharpness of the constant factorBC=|b(1− k)(δ + 2) + 2|+ |b(1− 2α+ k)(δ + 2) + 2|2[2|b(1− α)|+ |b(1− k)(δ + 2) + 2|+ |b(1− 2α+ k)(δ + 2) + 2|] ,64 E. A. Oyekan and T. O. Opoolawe consider the function:(2.8) f1(z) = z − 2|b(1− α)|Bz2 =Bz − 2|b(1− α)z2B(z ∈ U ; δ ≥ −1; 0 ≤ k ≤ 1; b ∈ C\{0}). Applying (1.10) with g(z) = z1− zand f(z) = f1(z) we have(2.9)Bz − 2b(α− 1)z2C≺ z1− z .Using the fact that(2.10) |Re z| ≤ |z|,we now show that(2.11) min{ReBz − 2b(α− 1)z2C: z ∈ U}= −12.Now,∣∣∣∣Re Bz − 2|b(1− α)|z2C∣∣∣∣ ≤∣∣∣∣Bz − 2|b(1− α)|z2C∣∣∣∣=|Bz − 2|b(1− α)|z2||C|≤ B|z|+ 2|b(1− α)||z2|C=B + 2|b(1− α)|C=12(2.12)(|z| = 1). This implies that(2.13)∣∣∣∣Re Bz − 2|b(1− α)|z2C∣∣∣∣ ≤ 12 ,i.e.,−12≤ Re Bz − 2|b(1− α)|z2C≤ 12.Hence,min{Re(BCf1(z)): z ∈ U}= −12,which completes the proof of Theorem 1.3.On a subordination result for analytic functions... 653. Some applications. Taking δ = 1 and b = 1 in Theorem 1.3, we obtainthe following:Corollary 1. If the function f(z) defined by (1.1) is in M δ∗(b, α), then(3.1)|5− 3α|2|6− 4α|(f ∗ g)(z) ≺ g(z)(z ∈ U ; α > 1, g ∈ K(α)). In particular,(3.2) Re(f(z)) > −|6− 4α||5− 3α| .The constant factor|5− 3α|2|6− 4α|cannot be replaced by any larger one.Remark 1. By taking α = 7145 > 1 in Corollary 1, we obtain the result ofAouf et al. [1]Taking b = 1, δ = 0 in Theorem 1.3, we obtain the following:Corollary 2. If the function f(z) defined by (1.1) is in M δ∗(b, α), then(3.3)|2− α||5− 3α|(f ∗ g)(z) ≺ g(z)(z ∈ U ; α > 1, g ∈ K(α)). In particular,(3.4) Re(f(z)) > −|5− 3α|2|2− α| , z ∈ U.The constant factor|2− α||5− 3α|cannot be replaced by any larger one.Remark 2. By taking α = 116 and α =2011 in Corollary 2, we obtain theresults of Selvaraj and Karthikeyan [4].Taking b = 1, δ = −1 and k = 0 in Theorem 1.3, we obtain the following:Corollary 3. If the function f(z) defined by (1.1) is in M δ∗(b, α), then(3.5)|3− α||8− 4α|(f ∗ g)(z) ≺ g(z)(z ∈ U ; α > 1, g ∈ K(α)). In particular,(3.6) Re(f(z)) > −|4− 2α||3− α| , z ∈ U.66 E. A. Oyekan and T. O. OpoolaThe constant factor|3− α||8− 4α|cannot be replaced by any larger one.Remark 3. If we take α = 7+3m3+m in Corollary 3, (m > 0) and in particularm = 1 (i.e., α = 52 > 1), we obtain the result of Attiya et al. [2].References[1] Aouf, M. K., Shamandy, A., Mostafa, A. O., El-Emam, F., Subordination resultsassociated with β-uniformly convex and starlike functions, Proc. Pakistan Acad. Sci.46, no. 2 (2009), 97–101.[2] Attiya, A. A., Cho, N. E., Kutbi, M. A., Subordination properties for certain analyticfunctions, Int. J. Math. Math. Sci. 2008, Article ID 63825 (2008).[3] Latha, S., Shivarudrappa, L., A note on coefficient estimates for a class of analyticfunctions, Advances in Inequalities for Series (2008), 143–149.[4] Selvaraj, C., Karthikeyan, K. R., Certain subordination results for a class of analyticfunctions defined by the generalized integral operator, Int. J. Comput. Math. Sci. 2,no. 4 (2008), 166–169.[5] Wilf, H. S., Subordinating factor sequences for some convex maps of unit circle, Proc.Amer. Math. Soc. 12 (1961), 689–693.E. A. Oyekan T. O. OpoolaDepartment of Mathematics and Statistics Department of MathematicsBowen University University of IlorinIwo, Osun State IlorinNigeria Nigeriae-mail: shalomfa@yahoo.com e-mail: opoolato@unilorin.edu.ngReceived August 8, 2011

On a subordination result for analytic functions defined by convolution

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On a subordination result for analytic functions defined by convolution

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On a subordination result for analytic functions defined by convolution

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