Univalence Conditions Related to a General Integral Operator
We consider a general integral operator based on two types of analytic functions, namely, regular functions and, respectively, functions having a positive real part. Some univalence conditions for this integral operator are obtained.
Let be the class of functions of the form which are analytic in the open unit disk, and normalized by . Also, let denote the subclass of consisting of univalent functions, in the open unit disk. We denote by , the class of functions which are analytic in , , for all and , the so-called Caratheodory functions.
In , Pescar introduced and studied the following integral operator: where are complex numbers, , , .
In this paper, we generalize this integral operator, by considering the general integral operator defined as follows: where , are complex numbers, , , , .
Remarks. This integral operator extends many other integral operators from related works on this field as those given by Srivastava, Mocanu, Owa, Pescar, Orhan, Breaz, and others. Using this new operator and the related univalence conditions that are going to be proved here, one can study some other already known operators in a unified perspective. Thus, for different particular cases of the parameters , , , our integral operator is shown to be an extension of the following integral operators.(i)For , the integral operator is the operator from (1.2), introduced by Pescar in .(ii)For , , , the integral operator was studied in , by Miller and Mocanu.(iii)For , , the integral operator was studied in , by Pescar and Breaz.(iv)For , , , and , the integral operator was studied in , by Breaz et al. and also in , by Srivastava et al.(v)For , , , we get the integral operator defined by D. Breaz and N. Breaz, in the paper .(vi)For , , , and , the operator was studied by Breaz et al., in the paper .
More precisely, if we are interested to study various properties of two or more different operators reminded above (and other similar operators which are not mentioned here), we can do this in an integrated manner by simply allowing the parameters involved in the definition, to be more general and consequently by studying only one operator, having the form (1.3). In this paper, we will study some univalence criteria for this new operator.
The following known results will be used in order to prove our results.
Lemma 1.1 (see ). Let be a complex number, and . If for all , then for any complex number , ,
Lemma 1.2 (see ). Let be a regular function in the open disk with , fixed. If has in one zero with multiplicity , then the equality for () can hold only if where is constant.
Lemma 1.3 (see ). For each ,
Lemma 1.4 (see ). If is regular in and , then
Also, for the statement of our main results, we need to define the following classes:
2. Main Results
Theorem 2.1. Let be complex numbers, , positive real numbers, , , and the functions , , .
If then the general integral operator , defined by (1.3), is in the class .
Proof. We consider the regular function
After some calculus, we have
for all .
Since , by applying Lemma 1.2 for , we get
Also since , from Lemma 1.2, we have
If we put these last two inequalities in (2.3), together with the inequality from the hypothesis, we get
Now we take into account the fact that thus obtaining
Further from Lemma 1.1, for , it is obvious that .
Corollary 2.2. Let , be complex numbers, , positive real numbers, , , . If then the function is in the class .
Proof. In Theorem 2.1, we take .
Corollary 2.3. Let , be complex numbers, , positive real numbers and , . If then the function belongs to the class .
Proof. In Theorem 2.1, we take .
Theorem 2.4. Let be complex numbers, , , and the functions , , , with
If then the general integral operator belongs to the class .
Proof. Let consider again the regular function:
After some calculus, we have
Since and , for all , from Lemma 1.2, we obtain
Using these last two inequalities, the inequality from the hypothesis, and the fact that from (2.16) we get for all .
Hence, by Lemma 1.1, we have that .
Corollary 2.5. Let be complex numbers, , and , , , with If then the integral operator defined by is in the class .
Proof. In Theorem 2.4, we take .
Corollary 2.6. Let be complex numbers, , , and , with If then the integral operator defined by is in the class .
Proof. We put in Theorem 2.4, .
Theorem 2.7. Let be complex numbers, , and , . If then the general integral operator .
Proof. We consider the same regular function as in the proof of the previous theorems, and after some calculus we get
for all .
Since , from Lemma 1.3, we get , , and since , from Lemma 1.4, we get , .
Now if we use (2.28) and (2.29) in (2.27), we obtain for all .
We consider two cases.
(1) If , we have and further if we use this inequality together with the inequality from the hypothesis, from (2.30), we get for all , and with Lemma 1.1 the proof is complete.
(2) If , we obtain and further if we put this last inequality in (2.30) we get for all .
Now by applying first the inequality condition from the hypothesis and then Lemma 1.1 for , the proof is complete.
Corollary 2.8. Let be complex numbers, , and . If then the integral operator is in the class .
Proof. In Theorem 2.7, we take .
Corollary 2.9. Let be complex numbers, , and . If then the integral operator is in the class .
Proof. In Theorem 2.7 we take .
The authors wish to thank the reviewers for their constructive comments which helped them to improve the quality of the paper.
V. Pescar, “On an integral operator,” Bulletin of the Transilvania University of Braşov, Series 3, vol. 4(53), no. 2, pp. 63–71, 2011.View at: Google Scholar
S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, vol. 225 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.
V. Pescar and V. D. Breaz, The Univalence of Integral Operators, Monograph, Marin Drinov Academic Publishing House, Sofia, Bulgaria, 2008.
D. Breaz, N. Breaz, and H. M. Srivastava, “An extension of the univalent condition for a family of integral operators,” Applied Mathematics Letters, vol. 22, no. 1, pp. 41–44, 2009.View at: Publisher Site | Google Scholar | Zentralblatt MATH
H. M. Srivastava, E. Deniz, and H. Orhan, “Some general univalence criteria for a family of integral operators,” Applied Mathematics and Computation, vol. 215, no. 10, pp. 3696–3701, 2010.View at: Publisher Site | Google Scholar | Zentralblatt MATH
D. Breaz and N. Breaz, “Two integral operators,” Studia Universitatis Babeş-Bolyai, Mathematica, vol. 47, no. 3, pp. 13–19, 2002.View at: Google Scholar | Zentralblatt MATH
D. Breaz, S. Owa, and N. Breaz, “A new integral univalent operator,” Acta Universitatis Apulensis, Mathematics, Informatics, no. 16, pp. 11–16, 2008.View at: Google Scholar | Zentralblatt MATH
N. N. Pascu, “An improvement of Becker's univalence criterion,” in Proceedings of the Commemorative Session: Simion Stoïlow (Braşov, 1987), pp. 43–48, University of Braşov, Braşov, Romania, 1987.View at: Google Scholar
O. Mayer, The Functions Theory of One Variable Complex, Bucuresti, Romania, 1981.
P. L. Duren, Univalent Functions—A Series of Comprehensive Studies in Mathematics, vol. 259 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, NY, USA, 1983.
P. T. Mocanu, T. Bulboacă, and G. Ş. Sălăgean, Teoria geometrică a funcţiilor univalente, Casa Cărţii de Ştiinţă, Cluj-Napoca, Romania, 2nd edition, 2006.