Univalence Conditions for a New Family of Integral Operators

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1 Filomat 30:5 (2016, DOI /FIL O Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: Univalence Conditions for a New Family of Integral Operators Adriana Oprea a, Daniel Breaz b, H. M. Srivastava c a Department of Mathematics, University of Piteşti, Street Târgul din Vale 1, R Piteşti, Argeş, România b Department of Mathematics, 1 Decembrie 1918 University of Alba Iulia, Street Nicolae Iorga 11-13, R Alba Iulia, Alba, România c Department of Mathematics Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada China Medical University, Taichung 40402, Taiwan, Republic of China Abstract. For analytic functions in the open unit disk U, we introduce a general family of integral operators. The main obect of the this paper is to present a systematic study if this general family of integral operators to determine the associated univalence conditions. Relevant connections of the results derived in this paper with those in several earlier works are also indicated. 1. Introduction, Definitions Preliminaries Let A be the class of functions f (z of the form: f (z = z + a k z k (z U, k=2 which are analytic in the open unit disk U = {z : z C z < 1} satisfy the following normalization conditions: f (0 = f (0 1 = 0. We denote by S the class of functions in A which are also univalent in U (see, for details, [4 [11. (1 A function f A is said to be in the class S (κ of starlike functions of order κ (0 κ < 1 in U if it satisfies the following inequality: ( z f (z R > κ (z U; 0 κ < 1. f (z 2010 Mathematics Subect Classification. Primary 30C45; Secondary 30C75 Keywords. Analytic functions; Univalent functions; Starlike functions; Integral operators; Regular functions; General Schwarz Lemma. Received: 01 April 2014; Accepted: 27 July 2014 Communicated by Dragan S. Dordević Research supported by the strategic proect PERFORM-POSDRU 159/1.5/S/ addresses: adriana_oprea@yahoo.com (Adriana Oprea, dbreaz@uab.ro (Daniel Breaz, harimsri@math.uvic.ca (H. M. Srivastava

2 Adriana Oprea et al. / Filomat 30:5 (2016, We denote by K(κ the class of convex functions of order κ (0 κ < 1 in U, that is, the class of functions in A which satisfy the following inequality: ( z f (z R f > κ (z U; 0 κ < 1. (z A function f A is said to belong to the class R(κ (0 < 1 if R[ f (z > κ (z U; 0 κ < 1. Recently, Frasin Jahangiri [6 studied the class B(, κ ( 0; 0 κ < 1, which consists of functions f A that satisfy the following condition: ( 1 z f (z f (z < 1 κ (z U; 0 κ < 1; 0. (2 This class B(, κ is a comprehensive class of normalized analytic functions in U that contains several other classes of analytic univalent functions in U such as B(1, κ =: S κ, B(0, κ =: R κ B(2, κ =: B(κ. In particular, the analytic univalent function class B(κ was studied by Frasin Darus [5. The problem of finding sufficient conditions for univalence of various integral operators has been investigated in many recent works (see, for example, [1 3, 9, 10, 12, 13; see also the other relevant references cited in each of these earlier works. Here, in our present investigation, we study the univalence conditions for the function I n, (z given by the following integral operator: I n, (z := 0 t 1 n [( ( f (t (t γ dt t t 1 when R( > 0 the functions f 1 (z,, f n (z 1 (z,, n (z are constrained suitably. We note here that the following theorems on univalence conditions of certain given integral operators were proven recently by Pascu [9 Pescar [10, respectively. Theorem 1. (see Pascu [9 Let f A C. If R( > 0 1 z 2R( z f (z R( f (z 1 (z U, (3 then the function F (z defined by F (z := ( 0 t 1 f (tdt 1 (z U, is in the class S of analytic univalent functions in U. Theorem 2. (see Pescar [10 Let c, α C with R(α > 0 c 1 (c 1. If the function f (z, regular in U, is given by (1 c z 2α + ( 1 z 2α z f (z α f (z 1 (z U, (4

3 then the function F α (z given by F α (z = ( α 0 t α 1 f (tdt Adriana Oprea et al. / Filomat 30:5 (2016, α = z + a k z k (z U is in the class S of regular univalent functions in U. k=2 In order to derive our main results, we recall here the General Schwarz Lemma as follows. General Schwarz Lemma (see, for example, [7 [8. Let the function f be regular in the disk U R = {z : z C z < R} with f (z < M ( z < UR ; M > 0 for fixed M > 0. If f has one zero at z = 0 with multiplicity m, then f (z M R m z m (z U R. (5 The equality in (5 holds true for z 0 only if where θ is a constant. f (z = e iθ M R m zm (z U R, 2. Univalence Conditions on the Class B(, α In this section, we first prove the univalence condition for the function I n, (z which is given in terms of the integral operator defined by (3. Theorem 3. Let the functions f, A ( = 1,, n. Suppose that, γ C, R( > 0 M, N 1 ( = 1,, n. Also let If R( [( (2 ( α M (2 + γ α N. (6 f, B(, α, 0 α < 1 0 ( = 1,, n f (z M (z N (z U; = 1,, n, then the function I n, (z given by the integral operator (3 is in the class S of analytic univalent functions in U. Proof. We begin by considering the function h(z defined by h(z := 0 n [( ( f (t (t γ dt (z U. (7 t t

4 Adriana Oprea et al. / Filomat 30:5 (2016, For this function h(z, which is regular in U, we calculate the first-order the second-order derivatives as follows: h (z = n [( ( f (z (z γ z z (8 h (z = z f (z f ( (z (z γ n [( ( γk fk (z k (z z z z + z 2 k=1 (k ( ( f (z (z γ z (z (z γ z z z 2 n [( ( γk fk (z k (z. (9 z z k=1 (k From (8 (9, we get h (z = z f (z z f (z 1 + γ (z (z 1, (10 which readily yields h (z z f (z f (z 1 + γ z (z (z 1. (11 Thus, clearly, we find from this last inequality (11 that 1 z 2R( R( h (z 1 z 2R( z f (z R( f (z + z γ (z (z 1 ( z 2R( R( f (z z f (z f (z z ( z 2R( γ R( (z z (z (z z. (12 By the hypothesis of Theorem 3, we have f (z M (z N (z U; = 1,, n. Therefore, by applying the General Schwarz Lemma to the functions f 1,, f n 1,, n, we obtain f (z M z (z N z (z U; = 1,, n. (13

5 Adriana Oprea et al. / Filomat 30:5 (2016, Now, by using the inequalities (2 (13, we get 1 z 2R( R( h (z 1 [( ( z 2R( R( f (z z 1 f (z z 2R( R( M 1 [( ( γ (z z 1 (z N 1 (14 which can be rewritten as follows: 1 z 2R( R( h (z 1 [(( 2 α M R( (z U, + γ (( 2 α N (15 (z U. If we make use of the condition (6 from the hypothesis of Theorem 3, this last inequality (15 yields 1 z 2R( R( h (z 1 (z U. (16 Finally, we apply Theorem 1 to the function h(z defined by (7. We thus conclude that the function I n, (z given by (3 is in the class S of analytic univalent functions in U. Corollary 1. Let the functions f (z ( = 1,, n (z ( = 1,, n be in the class A. Suppose also that, γ C ( = 1,, n with R( > 0 R( [( 3 α + γ ( 3 α. (17 If f, S (α ( = 1,, n for 0 α 1 ( = 1,, n f (z 1 (z 1 (z U; = 1,, n, then the function I n, (z given by (3 under that above constraints is in the class S of analytic univalent functions in U. Proof. Corollary 1 follows readily by setting in Theorem 3. = M = N = 1 ( = 1,, n Corollary 2. Let the functions f (z (z be in the class A. Suppose also that, γ C with R( > 0, M 1, N 1 R( ((2 αm 1 + γ ((2 αn 1. (18

6 Adriana Oprea et al. / Filomat 30:5 (2016, If f, B(, α (0 α < 1; 0 f (z M (z N (z U, then the function J (z given by J (z := [ 0 t 1 ( f (t t ( (t t γ dt 1 (19 is in the class S of analytic univalent functions in U. Proof. Since J (z = I 1, (z ( z U; R( > 0, which is an immediate consequence of the definitions (3 (19, Corollary 2 corresponds to the special case of Theorem 3 when n = 1. Next, by using Theorem 2 of Pescar [10, we get the following result. Theorem 4. Let the functions f, A ( = 1,, n. Suppose that Also let c,, γ C, R( > 0 M, N 1 ( = 1,, n. R( [( (2 ( α M (2 + γ α N (20 c 1 1 R( [( (2 ( α M (2 + γ α N. (21 If f, B(, α, 0 α < 1 0 ( = 1,, n f (z M (z N (z U; = 1,, n, then the function I n, (z given by the integral operator (3 is in the class S of analytic univalent functions in U. Proof. Just as in the proof of Theorem 3, we have h (z = z f (z z f (z 1 + γ (z (z 1 (z U, (22

7 Adriana Oprea et al. / Filomat 30:5 (2016, which, for a given constant c C, yields c z 2 + (1 z 2 zh (z h (z = c z 2 + (1 z 2 1 z f (z z f (z 1 + γ (z (z 1 c z f (z f (z + γ z (z (z c ( f (z z f (z f (z z ( γ (z z (z (z z (z U. (23 Now, from the hypothesis of Theorem 4, we have f (z M (z N (z U; = 1,, n. Applying the General Schwarz Lemma to the functions f 1,, f n 1,, n, we obtain f (z M z (z N z (z U; = 1,, n, (24 which, in conunction with the inequality (2, leads us to following result: c z 2 + (1 z 2 zh (z h (z c [( ( f (z z 1 f (z M 1 [( ( γ (z z 1 (z N 1 c R( [(( 2 α M + γ (( 2 α N (25 (z U. Thus, from the condition (21 of Theorem 4, we find that c z 2 + (1 z 2 zh (z h (z 1 (z U. (26 Finally, by applying Theorem 2 to the function h(z given by (7, we deduce the desired assertion that the function I n, (z given by the integral operator (3 is in the class S of analytic univalent functions in U. Corollary 3. Let the functions f (z ( = 1,, n (z ( = 1,, n be in the class A. Suppose also that c,, γ C ( = 1,, n with R( > 0 R( [( 3 α + γ ( 3 α. (27

8 Adriana Oprea et al. / Filomat 30:5 (2016, If f, S (α ( = 1,, n for 0 α 1 ( = 1,, n, f (z 1 (z 1 (z U; = 1,, n c 1 1 R( [( 3 α + γ ( 3 α, (28 then the function I n, (z given by (3 under the above constraints is in the class S of analytic univalent functions in U. Proof. Corollary 3 follows easily upon setting in Theorem 4. = M = N = 1 ( = 1,, n Corollary 4. Let the functions f (z (z be in the class A. Suppose also that c,, γ C with R( > 0, M 1, N 1 R( ((2 αm 1 + γ ((2 αn 1. (29 If f, B(, α (0 α < 1; 0, f (z M (z N (z U c 1 1 [( 2 αm 1 + γ (2 αn 1, (30 R( then the function J (z given by (19 is in the class S of analytic univalent functions in U. Proof. In its special case when n = 1, Theorem 4 would obviously correspond to Corollary Concluding Remarks Observations Our present investigation was motivated essentially by several recent works dealing with the interesting problem of finding sufficient conditions for univalence of normalized analytic functions which are defined in terms of various families integral operators (see, for example, [1 3, 9, 10, 12, 13; see also the other relevant references cited in each of these earlier works. In our study here, we have successfully determined the univalence conditions for the function I n, (z given by the general family of integral operators in (3. Our main results (Theorems 3 4 in this paper are shown to yield several corollaries consequences. Some of these applications of our main results are stated here as Corollaries 1, 2, 3 4. Derivations of further corollaries consequences of the results presented in this paper, including also their connections with known results given in several earlier works, are being left here as exercises for the interested reader. Acknowledgements This work was supported by the strategic proect PERFORM (POSDRU 159/1.5/S/ inside POSDRU Romania 2014, co-financed by the European Social Fund-Investing in People.

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