The main object of the present paper is to extend the univalence condition for a family of integral operators. Relevant connections of some of the results obtained in this paper with those in earlier works are also provided

Deniz, Erhan

Orhan, Halit

English

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

doi: 10.2478/v10062-010-0012-zANNALESU 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. LXIV, NO. 2, 2010 SECTIO A 29–35ERHAN DENI˙Z and HALIT ORHANAn extension of the univalence criterionfor a family of integral operatorsAbstract. The main object of the present paper is to extend the univalencecondition for a family of integral operators. Relevant connections of some ofthe results obtained in this paper with those in earlier works are also provided.1. Introduction and preliminaries. Let A denote the class of functionsf normalized by(1.1) f(z) = z +∞∑n=2anzn,which are analytic in the open unit disk U = {z : z ∈ C and |z| < 1} andsatisfy the conditions f(0) = f ′(0)− 1 = 0.Consider S = {f ∈ A : f is a univalent function in U}.A function f(z) ∈ A is said to be a member of the class B(γ) if and onlyif(1.2)∣∣∣∣z2f ′(z)f2(z) − 1∣∣∣∣ < 1− γ, 0 ≤ γ < 1.Recently, Frasin and Darus (see [6]) defined and studied the class B(γ).In his paper Frasin (see [4]) obtained some results for functions belonging2000 Mathematics Subject Classification. 30C45.Key words and phrases. Integral operator, analytic functions, univalent functions, openunit disk, univalence criterion.30 E. Deniz and H. Orhanto this class and also he showed that if f(z) ∈ B(γ) then f(z) satisfies thefollowing inequality(1.3)∣∣∣∣zf ′′(z)f ′(z)∣∣∣∣ ≤ (1− γ)(2 + |z|)1− |z| (z ∈ U).For γ = 0 the class B(0) = T was studied by Ozaki and Nunokawa(see [8]).We denote byW the class of functions w which are analytic in U satisfyingthe conditions |w(z)| < 1 and w(0) = w′(0) = 0 for all z ∈ U .Now, by Schwarz’s lemma, it follows that(1.4) |w(z)| < |z| .In [7], we see that if w(z) ∈ W, then w(z) satisfies(1.5)∣∣w′(z)∣∣ ≤ 1− |w(z)|21− |z|2 (z ∈ U).In [11], N. Seenivasagan and D. Breaz considered the following family ofintegral operators Fα1,α2,...,αn,β(z) defined as follows(1.6) Fα1,α2,...,αn,β(z) :=β z∫0tβ−1n∏i=1(fi(t)t) 1αidt 1β ,where fi ∈ A, f ′′i (0) = 0 and α1, α2, . . . , αn, β ∈ C for all i ∈ {1, 2, . . . , n}.When αi = α for all i ∈ {1, 2, . . . , n}, Fα1,α2,...,αn,β(z) becomes the inte-gral operator Fα,β(z) considered in (see [1]).We begin by recalling each of the following theorems dealing with univa-lence criterion, which will be required in our present paper.In [10], Pascu proved the following theorem.Theorem 1 (Pascu [10]). Let β be a complex number with Re(β) > 0 andf ∈ A. If1− |z|2Re(β)Re(β)∣∣∣∣zf ′′(z)f ′(z)∣∣∣∣ ≤ 1,for z ∈ U , then the functionFβ(z) :=β z∫0tβ−1f ′(t)dt 1βbelongs to S.In [9], Pascu and Pescar obtained the next result.An extension of the univalence criterion for a family... 31Theorem 2 (Pascu and Pescar [9]). Let β and µ be complex numbers, andg ∈ S. If Re(β) > 0 and |µ| ≤ min{12 Re(β); 14} , then the functionGβ,µ(z) :=β z∫0tβ−1(g(t)t)µdt 1βbelongs to S.Note that Theorem 2 includes the special case of Pascu and Pescar’stheorem (see [9]) when Re(α) = Re(β).In the present paper, we propose to investigate further univalence condi-tion involving the general a family of integral operators defined by (1.6).2. Main results. In this section, we first state an inclusion for f(z) ∈B(γ), then we give the main univalence condition involving the general in-tegral operator given by (1.6).Theorem 3. If f(z) ∈ B(γ), then the inequality is satisfied(2.1)∣∣∣∣zf ′(z)f(z) − 1∣∣∣∣ ≤ (1− γ)(1 + |z|)1− |z|for all z ∈ U .Proof. Let f(z) ∈ B(γ). Then we have(2.2)z2f ′(z)f2(z)= 1 + (1− γ)w(z),where w(z) ∈ W. By applying the logarithmic differentiation, we obtainfrom (2.2) thatzf ′′(z)f ′(z)=(1− γ)zw′(z)1 + (1− γ)w(z) +2zf ′(z)f(z)− 2andzf ′(z)f(z)− 1 = 12(zf ′′(z)f ′(z)− (1− γ)zw′(z)1 + (1− γ)w(z)),thereby, it follows that∣∣∣∣zf ′(z)f(z) − 1∣∣∣∣ = ∣∣∣∣12(zf ′′(z)f ′(z)− (1− γ)zw′(z)1 + (1− γ)w(z))∣∣∣∣≤ 12(∣∣∣∣zf ′′(z)f ′(z)∣∣∣∣+ ∣∣∣∣ (1− γ)zw′(z)1 + (1− γ)w(z)∣∣∣∣)≤ 12(∣∣∣∣zf ′′(z)f ′(z)∣∣∣∣+ (1− γ) |z| |w′(z)|1− (1− γ) |w(z)|).32 E. Deniz and H. OrhanFrom (1.3) and (1.5), we have(2.3)∣∣∣∣zf ′(z)f(z) − 1∣∣∣∣ ≤ 12((1− γ)(2 + |z|)1− |z| +(1− γ) |z|1− (1− γ) |w(z)|1− |w(z)|21− |z|2)and for 0 ≤ γ < 1, it is easy to show that(2.4)1− |w(z)|1− (1− γ) |w(z)| ≤ 1 (z ∈ U).From (1.4), (2.3) and (2.4), we obtain that(2.5)∣∣∣∣zf ′(z)f(z) − 1∣∣∣∣ ≤ (1− γ)(1 + |z|)1− |z| .This evidently completes the proof of Theorem 3. Next we prove the following main theorem.Theorem 4. Let fi(z) ∈ B(γ) for i ∈ {1, 2, . . . , n}. Let β be a complexnumber with Re(β) > 0. If(2.6)n∑i=11|αi| ≤ min{12(1− γ) Reβ;14(1− γ)}for all z ∈ U , then the functionFα1,α2,...,αn,β(z) :=β z∫0tβ−1n∏i=1(fi(t)t) 1αidt 1βbelongs to S.Proof. Define functionh(z) =z∫0n∏i=1(fi(t)t) 1αidt.We have h(0) = h′(0)− 1 = 0. Also, a simple computation yields(2.7) h′(z) =n∏i=1(fi(z)z) 1αi.Making use of logarithmic differentiation in (2.7), we obtain(2.8)zh′′(z)h′(z)=n∑i=11αi(zf ′i(z)fi(z)− 1).We thus have from (2.8) that∣∣∣∣zh′′(z)h′(z)∣∣∣∣ ≤ n∑i=11|αi|∣∣∣∣zf ′i(z)fi(z) − 1∣∣∣∣.An extension of the univalence criterion for a family... 33By using the Theorem 3, we get the inequality(2.9)∣∣∣∣zh′′(z)h′(z)∣∣∣∣ ≤ n∑i=11|αi|(1− γ)(1 + |z|)1− |z| .From (2.9), we obtain(2.10)1− |z|2Re(β)Re(β)∣∣∣∣zh′′(z)h′(z)∣∣∣∣ ≤ 1− |z|2Re(β)Re(β) (1− γ)(1 + |z|)1− |z|n∑i=11|αi|≤ 1− |z|2Re(β)1− |z|2(1− γ)Re(β)n∑i=11|αi|for all z ∈ U .Let us denote |z| = x, x ∈ [0, 1), Re(β) = a > 0 and ψ(x) = 1−x2a1−x . It iseasy to prove that(2.11) ψ(x) ≤{1, if 0 < a < 122a, if 12 < a <∞.From (2.6), (2.10) and (2.11), we have1− |z|2Re(β)Re(β)∣∣∣∣zh′′(z)h′(z)∣∣∣∣ ≤2(1−γ)Re(β)n∑i=11|αi| , if 0 < Re(β) <124(1− γ)n∑i=11|αi| , if12 < Re(β) <∞≤ 1for all z ∈ U .Finally, by applying Theorem 1, we conclude that the functionFα1,α2,...,αn,β(z) defined by (1.6) is in the function class S. This evidentlycompletes the proof of Theorem 4. 3. Some applications of Theorem 4. In this section, we give some re-sults of Theorem 4.First of all, upon setting αi = α, for all i ∈ {1, 2, . . . , n} in Theorem 4,we immediately arrive at the following application of Theorem 4.Corollary 1. Let fi(z) ∈ B(γ) for i ∈ {1, 2, . . . , n}. Let β be a complexnumber with Re(β) > 0. If(3.1)1|α| ≤ min{12n(1− γ) Reβ;14n(1− γ)}holds for all z ∈ U , then the functionFα,β(z) :=β z∫0tβ−1n∏i=1(fi(t)t) 1αdt 1β34 E. Deniz and H. Orhanbelongs to S.Next we set n = 1 in Theorem 4, we thus obtain the following interestingconsequence of Theorem 4.Corollary 2. Let the functions f(z) ∈ B(γ). Let β be a complex numberwith Reβ > 0. If(3.2)1|α| ≤ min{12(1− γ) Reβ;14(1− γ)}holds for all z ∈ U , then the functionGβ,α(z) :=β z∫0tβ−1(f(t)t) 1αdt 1βbelongs to S.Remark 1.(i) Corollary 2 provides an extension of Theorem 2 due to Pascu andPescar (see [9]).(ii) If we set γ = 0, n = 1 and 1α = µ in Theorem 4, we obtain Theorem2 due to Pascu and Pescar (see [9]).(iii) If we put γ = 0, β = 1 and α instead of 1α in Corollary 2, we arriveat the result by Kim and Merkes (see [5]).Remark 2. Some authors gave similar univalence conditions by usingbounded functions f(z) ∈ A in their papers, see the works (for exampleBreaz et al. (see [2]), Breaz et al. (see [3])). We note that the functionsf ∈ A do not have to be bounded.Acknowledgement. Authors would like to thank the referee for thoughtfulcomments and suggestions.References[1] Breaz, D., Breaz, N., The univalent conditions for an integral operator on the classesS(p) and T2, J. Approx. Theory Appl. 1(2) (2005), 93–98.[2] Breaz, D., Breaz, N., Sufficient univalence conditions for analytic functions, J. In-equal. Appl. 2007 (2007), Article ID 86493, 5 pp.[3] Breaz, D., Breaz, N. and Srivastava, H. M., An extension of the univalent conditionfor a family of integral operators, Appl. Math. Lett. 22 (2009), 41–44.[4] Frasin, B. A., A note on certain analytic and univalent functions, Southeast AsianBull. Math. 28(5) (2004), 829–836.[5] Kim, Y. J., Merkes, E. P., On an integral of powers of a spirallike function, Kyung-pook Math. J. 12 (1972), 249–252.[6] Frasin, B. A., Darus, M., On certain analytic univalent functions, Int. J. Math. Math.Sci. 25(5) (2001), 305–310.[7] Nehari, Z., Conformal Mapping, Dover Publications, Inc., New York, 1975.An extension of the univalence criterion for a family... 35[8] Ozaki, S., Nunokawa, M., The Schwarzian derivative and univalent functions, Proc.Amer. Math. Soc. 33 (1972), 392–394.[9] Pascu, N. N., Pescar, V., On the integral operators of Kim–Merkes and of Pfaltzgraff,Mathematica (Cluj) 32(55), no. 2 (1990) 185–192.[10] Pascu, N., An improvement of Becker’s univalence criterion, Proceedings of the Com-memorative Session Simion Stoilow (Bras¸ov, 1987), 43–98, Univ. Bras¸ov, Bras¸ov,1987.[11] Seenivasagan, N., Breaz, D., Certain sufficient conditions for univalence, Gen. Math.15(4) (2007), 7–15.E. Deniz H. OrhanDepartment of Mathematics Department of MathematicsFaculty of Science Faculty of ScienceAtaturk University Ataturk UniversityErzurum, 25240 Erzurum, 25240Turkey Turkeye-mail: edeniz@atauni.edu.tr e-mail: horhan@atauni.edu.trReceived October 29, 2009

An extension of the univalence criterion for a family of integral operators

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An extension of the univalence criterion for a family of integral operators

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