EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 6, No. 3, 2013, 307-314 ISSN 1307-5543 – www.ejpam.com Some Properties for Certain General Integral Operator Vasile Marius Macarie1,∗, Daniel Breaz 2 1 Department of Mathematics and Computer Sciences, University of Pitȩsti, Pitȩsti, România 2 Department of Mathematics, "1 Decembrie 1918" University of Alba Iulia, Alba Iulia, România Abstract. In this paper we consider some subclasses of the class of analytic functions defined in the open unit disk of the complex plane and we study some properties for an integral operator on these classes. Particular results are presented. 2010 Mathematics Subject Classifications: 30C45 Key Words and Phrases: integral operator, analytic function, convex function, starlike function 1. Introduction LetA denote the class of the functions f of the form f (z) = z+ ∞ ∑ n=2 anzn which are analytic in the open unit disk U = {z ∈ C : |z| < 1}. We also denote by S the subclass ofA consisting of functions which are univalent in U . A function f ∈A is said to be convex of order α, 0≤ α < 1 if it satisfies the condition Re � z f ′′(z) f ′(z) + 1 � > α, (z ∈ U) and we denote this class by K(α). A function f ∈A is said to be starlike of order α, 0≤ α≤ 1 if it satisfies the condition Re � z f ′(z) f (z) � > α, (z ∈ U) and denote this class by S∗(α). ∗Corresponding author. Email addresses: macariem@yahoo.com (V. Macarie), dbreaz@uab.ro (D. Breaz) http://www.ejpam.com 307 c© 2013 EJPAM All rights reserved. V. Macarie, D. Breaz / Eur. J. Pure Appl. Math, 6 (2013), 307-314 308 Let N (ρ) be the subclass ofA consisting of the functions f which satisfy the inequality Re � 1+ z f ′′(z) f ′(z) � < ρ, ρ > 1, (z ∈ U). This class was studied by S. Owa and H.M. Srivastava in [5]. A. Mohammed et al. considered in [4] MT (µ,β) the subclass of A consisting of the functions f which satisfy the inequality � � � � z f ′(z) f (z) − 1 � � � � < β � � � � µ z f ′(z) f (z) + 1 � � � � , 0< β ≤ 1, 0≤ µ < 1, (z ∈ U). Also, Frasin and Jahangiri introduced in [2] the family B(µ,α), µ ≥ 0, 0 ≤ α ≤ 1, consisting of the functions f which satisfy the condition � � � � f ′(z) � z f (z) �µ − 1 � � � � < 1−α, (z ∈ U). This family is a comprehensive class of analytic functions that includes various classes of analytic functions. We haveB(1,α)≡ S∗(α) andB(0,α)≡ R(α). Let β−Sp(α) be the subclass ofA consisting of the functions f which satisfy the inequality Re � z f ′(z) f (z) −α � ≥ β � � � � z f ′(z) f (z) − 1 � � � � , −1≤ α≤ 1, β > 0, (z ∈ U). This class was studied by M. Darus in [1]. A function f is said to be in the class KD(µ,α) if it satisfies the inequality Re � z f ′′(z) f ′(z) + 1 � ≥ µ � � � � z f ′′(z) f ′(z) � � � � +α, µ≥ 0, 0≤ α < 1, (z ∈ U). This class was studied by S. Shams et al. in [6]. In the present paper we study some properties for the integral operator Gn defined by Gn(z) = ∫ z 0 n ∏ i=1 � fi(t) �γi−1 (g ′i(t)) ηi dt (1) on the classes presented above. In order to prove our main results we need the following lemma: Lemma 1 (General Schwarz Lemma [3]). Let the function f be regular in the disk UR = {z ∈ C : |z| < R}, with | f (z)| < M for fixed M. If f has one zero with multiplicity order bigger than m for z = 0, then | f (z)| ≤ M Rm · |z| m (z ∈ UR). The equality can hold only if f (z) = eiθ · M Rm · z m, where θ is constant. V. Macarie, D. Breaz / Eur. J. Pure Appl. Math, 6 (2013), 307-314 309 2. Main Results Theorem 1. Let γi ∈ R, γi > 1, ηi ∈ R, ηi > 0 for all i = 1, 2, . . . , n, the functions fi ∈ MT (µi ,βi), 0 < βi ≤ 1, 0 ≤ µi < 1 and gi ∈ A for all i = 1,2, . . . , n satisfying the conditions � � � � � f ′i (z) fi(z) � � � � � < Mi , (Mi ≥ 1) for all i = 1,2, . . . , n (2) and � � � � � g ′′i (z) g ′i(z) � � � � � < Ni , (Ni ≥ 1) for all i = 1, 2, . . . , n. (3) Then the integral operator Gn defined in (1) is in N (ρ), where ρ = 1+ n ∑ i=1 � (γi − 1)(βiµi Mi + βi + 1) +ηiNi � Proof. From (1), we have G′n(z) = n ∏ i=1 � fi(z) �γi−1 (g ′i(z)) ηi and zG′′n (z) G′n(z) = n ∑ i=1 (γi − 1) z f ′i (z) fi(z) + n ∑ i=1 ηi zg ′′i (z) g ′i(z) . Thus, we have Re � zG′′n (z) G′n(z) + 1 � = n ∑ i=1 (γi − 1)Re � z f ′i (z) fi(z) � + n ∑ i=1 ηiRe � zg ′′i (z) g ′i(z) � + 1. Since Re w ≤ |w|, then Re � zG′′n (z) G′n(z) + 1 � ≤ n ∑ i=1 (γi − 1) � � � � � z f ′i (z) fi(z) � � � � � + n ∑ i=1 ηi � � � � � zg ′′i (z) g ′i(z) � � � � � + 1. (4) Using that fi ∈MT (µi ,βi) for all i = 1, 2, . . . , n in relation (4), we obtain Re � zG′′n (z) G′n(z) + 1 � ≤ n ∑ i=1 (γi − 1) � � � � � z f ′i (z) fi(z) − 1 � � � � � + 1 ! + n ∑ i=1 ηi � � � � � zg ′′i (z) g ′i(z) � � � � � + 1 < n ∑ i=1 (γi − 1)βi � � � � � µi z f ′i (z) fi(z) + 1 � � � � � + n ∑ i=1 (γi − 1) + n ∑ i=1 ηi � � � � � zg ′′i (z) g ′i(z) � � � � � + 1 V. Macarie, D. Breaz / Eur. J. Pure Appl. Math, 6 (2013), 307-314 310 and using the hypothesis (2) and (3) in this last relation we have Re � zG′′n (z) G′n(z) + 1 � < n ∑ i=1 � (γi − 1)(βiµi Mi + βi + 1) +ηiNi � + 1= ρ This completes the proof of our theorem. Letting n = 1, γ1 = γ, η1 = η, M1 = M , N1 = N , µ1 = µ, β1 = β , f1 = f and g1 = g in Theorem 1, we have Corollary 1. Let γ ∈ R, γ > 1, η ∈ R, η > 0, the functions f ∈ MT (µ,β), 0 < β ≤ 1, 0≤ µ < 1 and g ∈A satisfying the conditions � � � � f ′(z) f (z) � � � � < M , (M ≥ 1) and � � � � g ′′(z) g ′(z) � � � � < N , (N ≥ 1). Then the integral operator G1(z) = ∫ z 0 � f (t) �γ−1 (g ′(t))ηdt is in N (ρ), where ρ = (γ− 1)(βµM + β + 1) +ηN + 1. Letting γ= 2, η= 1 in Corollary 1, we have Corollary 2. Let f ∈MT (µ,β), 0< β ≤ 1, 0≤ µ < 1 and g ∈A satisfying the conditions � � � � f ′(z) f (z) � � � � < M , (M ≥ 1) and � � � � g ′′(z) g ′(z) � � � � < N , (N ≥ 1). Then the integral operator G(z) = ∫ z 0 f (t)g ′(t)dt is in N (ρ), where ρ = β(µM + 1) + N + 2. Theorem 2. Let γi ∈ R, γi > 1, ηi ∈ R, ηi > 0 for all i = 1, 2, . . . , n, the functions fi ∈ B(µi ,αi), µi ≥ 0, 0 ≤ αi < 1 satisfying the conditions | fi(z)| ≤ Mi , (Mi ≥ 1) and gi ∈ N (ρi), ρi > 1 for all i = 1, 2, . . . , n. If n ∑ i=1 h (γi − 1)(2−αi)M µi−1 i +ηi(ρi − 1) i < 1, then the integral operator Gn defined in (1) is in K(δ), where δ = 1− n ∑ i=1 h (γi − 1)(2−αi)M µi−1 i +ηi(ρi − 1) i . V. Macarie, D. Breaz / Eur. J. Pure Appl. Math, 6 (2013), 307-314 311 Proof. From (1), we have G′n(z) = n ∏ i=1 � fi(z) �γi−1 (g ′i(z)) ηi and zG′′n (z) G′n(z) = n ∑ i=1 (γi − 1) z f ′i (z) fi(z) + n ∑ i=1 ηi zg ′′i (z) g ′i(z) . Hence � � � � zG′′n (z) G′n(z) � � � � ≤ n ∑ i=1 (γi − 1) � � � � � f ′i (z) � z fi(z) �µi − 1 � � � � + 1 � � � � � fi(z) z � � � � µi−1 + n ∑ i=1 ηi � � � � � zg ′′i (z) g ′i(z) + 1 � � � � � − 1 ! (5) Since | fi(z)| ≤ Mi for all i = 1, 2, . . . , n, applying the General Schwarz Lemma, it results � � � � fi(z) z � � � � ≤ Mi for all i = 1,2, . . . , n. (6) From (5) and (6), using that fi ∈B(µi ,αi) and gi ∈ N (ρi) for all i = 1,2, . . . , n, we obtain � � � � zG′′n (z) G′n(z) � � � � < n ∑ i=1 h (γi − 1)(2−αi)M µi−1 i +ηi(ρi − 1) i = 1−δ This completes the proof of our theorem. Letting n = 1, γ1 = γ, η1 = η, M1 = M , µ1 = µ, α1 = α, ρ1 = ρ, f1 = f and g1 = g in Theorem 2, we have Corollary 3. Let γ ∈ R, γ > 1, η ∈ R, η > 0, the functions f ∈ B(µ,α), µ ≥ 0, 0 ≤ α < 1, satisfying the condition | f (z)| ≤ M, (M ≥ 1) and g ∈ N (ρ), ρ > 1. If (γ− 1)(2−α)Mµ−1+η(ρ− 1)< 1 then the integral operator G1(z) = ∫ z 0 � f (t) �γ−1 (g ′(t))ηdt is in K(δ), where δ = 1+ (γ− 1)(α− 2)Mµ−1+η(1−ρ). Letting µi = 0 and Mi = M for all i = 1,2, . . . , n in Theorem 2, we have V. Macarie, D. Breaz / Eur. J. Pure Appl. Math, 6 (2013), 307-314 312 Corollary 4. Let γi ∈ R, γi > 1, ηi ∈ R, ηi > 0 for all i = 1, 2, . . . , n, the functions fi ∈ R(αi), 0 ≤ αi < 1, satisfying the conditions | fi(z)| ≤ M, (M ≥ 1) and gi ∈ N (ρi), ρi > 1 for all i = 1, 2, . . . , n. If n ∑ i=1 � (γi − 1)(2−αi) 1 M +ηi(ρi − 1) � < 1 then the integral operator Gn defined in (1) is in K(δ), where δ = 1− n ∑ i=1 � (γi − 1)(2−αi) 1 M +ηi(ρi − 1) � . Letting µi = 1 and Mi = M for all i = 1, 2, . . . , n in Theorem 2, we have Corollary 5. Let γi ∈ R, γi > 1, ηi ∈ R, ηi > 0 for all i = 1, 2, . . . , n, the functions fi ∈ S∗(αi), 0 ≤ αi < 1, satisfying the conditions | fi(z)| ≤ M, (M ≥ 1) and gi ∈ N (ρi), ρi > 1 for all i = 1,2, . . . , n. If n ∑ i=1 � (γi − 1)(2−αi) +ηi(ρi − 1) � < 1 then the integral operator Gn defined in (1) is in K(δ), where δ = 1− n ∑ i=1 � (γi − 1)(2−αi) +ηi(ρi − 1) � . Theorem 3. Let γi ∈ R, γi > 1, ηi ∈ R, ηi > 0 for all i = 1, 2, . . . , n, the functions fi ∈ ρi − Sp(εi), −1 ≤ εi ≤ 1, ρi > 0 and gi ∈ KD(µi ,αi), 0 ≤ αi < 1, µi ≥ 0 for all i = 1,2, . . . , n. If 0< n ∑ i=1 � (1− γi)εi +ηi(1−αi) � ≤ 1 then the integral operator Gn defined in (1) is in K(δ), where δ = 1+ n ∑ i=1 � (γi − 1)εi +ηi(αi − 1) � . Proof. Following the same steps as in Theorem 1, we obtain that zG′′n (z) G′n(z) = n ∑ i=1 (γi − 1) z f ′i (z) fi(z) + n ∑ i=1 ηi zg ′′i (z) g ′i(z) and hence zG′′n (z) G′n(z) + 1= n ∑ i=1 � (γi − 1) � z f ′i (z) fi(z) − εi � + (γi − 1)εi � + n ∑ i=1 � ηi � zg ′′i (z) g ′i(z) + 1 � −ηi � + 1. V. Macarie, D. Breaz / Eur. J. Pure Appl. Math, 6 (2013), 307-314 313 We calculate the real part from both terms of the above expression and obtain Re � zG′′n (z) G′n(z) + 1 � = n ∑ i=1 (γi − 1)Re � z f ′i (z) fi(z) − εi � + n ∑ i=1 (γi − 1)εi + n ∑ i=1 ηiRe � zg ′′i (z) g ′i(z) + 1 � − n ∑ i=1 ηi + 1. (7) From (7), using that fi ∈ ρi − Sp(εi) and gi ∈ KD(µi ,αi) for all i = 1,2, . . . , n, we have Re � zG′′n (z) G′n(z) + 1 � ≥ n ∑ i=1 (γi − 1)ρi � � � � � z f ′i (z) fi(z) − 1 � � � � � + n ∑ i=1 (γi − 1)εi + n ∑ i=1 ηi µi � � � � � zg ′′i (z) g ′i(z) � � � � � +αi ! − n ∑ i=1 ηi + 1. Then Re � zG′′n (z) G′n(z) + 1 � ≥ n ∑ i=1 (γi − 1)ρi � � � � � z f ′i (z) fi(z) − 1 � � � � � + n ∑ i=1 (γi − 1)εi + n ∑ i=1 ηiµi � � � � � zg ′′i (z) g ′i(z) � � � � � + n ∑ i=1 ηi � αi − 1 � + 1. (8) Since (γi − 1)ρi � � � � � z f ′i (z) fi(z) − 1 � � � � � > 0 and ηiµi � � � � � zg ′′i (z) g ′i(z) � � � � � ≥ 0 for all i = 1,2, . . . , n, we obtain from (8) that Re � zG′′n (z) G′n(z) + 1 � > n ∑ i=1 � (γi − 1)εi +ηi(αi − 1) � + 1= δ. This completes the proof of our theorem. Letting n = 1, γ1 = γ, η1 = η, ρ1 = ρ, ε1 = ε, µ1 = µ, α1 = α, f1 = f and g1 = g in Theorem 3, we have Corollary 6. Let γ ∈ R, γ > 1, η ∈ R, η > 0, the functions f ∈ ρ − Sp(ε), −1 ≤ ε ≤ 1, ρ > 0 and g ∈ KD(µ,α), 0≤ α < 1, µ≥ 0. If 0< (1− γ)ε+η(1−α)≤ 1 then the integral operator G1(z) = ∫ z 0 � f (t) �γ−1 (g ′(t))ηdt is in K(δ), where δ = 1+ (γ− 1)ε+η(α− 1). Letting γ= 2 and η= 1 in Corollary 6, we have REFERENCES 314 Corollary 7. Let the functions f ∈ ρ−Sp(ε), −1≤ ε ≤ 1, ρ > 0 and g ∈ KD(µ,α), 0≤ α < 1, µ≥ 0. If 0< 1−α− ε ≤ 1 then the integral operator G(z) = ∫ z 0 f (t)g ′(t)dt is in K(δ), where δ = ε+α. References [1] M Darus. Certain class of uniformly analytic functions. Acta Mathematica Academiae Pedagogicae Nyregyhaziensis, 24:354-358, 2008. [2] B A Frasin and J M Jahangiri. A new and comprehensive class of analytic functions. Annals of Oradea University-Mathematics Fascicola, XV:59-62, 2008. [3] O Mayer. The functions theory of one variable complex. Bucuresti, 1981. [4] A Mohammed, M Darus, and D Breaz, Some properties for certain integral operators. Acta Universitatis Apulensis, 23:79-89, 2010. [5] S Owa and H M Srivastava. Some generalized convolution properties associated with certain subclasses of analytic functions. Journal of Inequalities in Pure and Applied Math- ematics, 3(3):42:1-13, 2003. [6] S Shams, S R Kulkarni, and J M Jahangiri. Classes of uniformly starlike and convex func- tions. International Journal of Mathematics and Mathematical Sciences, 55:2959-2961, 2004.