Abstract

In this paper, we consider a new general integral operator, defined as a joint extension of two already known integral operators, and prove some univalence properties for this operator. Some other well-known operators are mentioned as particular cases of our general operator, and known results are outlined also as particular cases of our results.

1. Introduction

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 .

One of the topics in geometric function theory is the study of geometric properties of the integral operators. Such properties like univalence, starlikeness, and convexity of different integral operators on various classes of analytic functions have been studied by many authors in their works, among which we mention the papers [1–13].

In our habilitation thesis [2], we introduced (as one of further research directions), the general integral operator , with and as follows: where are positive real numbers, is a complex number with , , , is exponential function, and is the Hadamard product.

This operator is designed as a joint extension of two already known operators given by Attiya [1] and Frasin [8], but it also covers other known integral operators.

Remark 1. (i)For , we obtain the integral operator given by Attiya in [1](ii)For , we get the integral operator defined by Frasin in [8], with positive real numbers, for (iii)For , and , , we get the integral operator, defined by Breaz D. and Breaz N. in [3], with positive real numbers, for (iv)For , , , and , we get the integral operator, defined by Breaz D. and Breaz N. in [3], with positive real numbers, for (v)For , , , and , we get the integral operator, defined by Breaz et al. in [4], with positive real numbers, for (vi)For , , , and , we get the Alexander integral operator, (vii)For , , , and , we get the Miller-Mocanu integral operator, , with positive real number

In that follows, we study the univalence of this general integral operator. The following known results will be used in order to prove our main results.

Lemma 2 (see [14]). Let be a complex number, , and . If for all , then

Lemma 3 (see [15]). Let be a complex number, , and . If for all , then for any complex number , with , we have

Lemma 4 (see [16]). Let be a complex number, , with . If satisfies for all , then

Lemma 5 (see [10]). Let and be complex numbers, , , and . If for all and for some , , then for any complex number , with , we have

Lemma 6 (Schwarz) (see [17]). 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.

In order to make the statement for the main results, we need to define the following class (a different class but using the same idea was defined by Frasin in [8]):

2. Main Results

Theorem 7. Let be , positive real numbers, , complex number, , and . If , , then .

Proof. For the beginning, let consider the function: We have , where By logarithmical differentiation, we get and further Since , according to Schwarz lemma, we obtain Further, since , if we put (19) in (18), we have On the other hand, the maximum value of the function is , obtained for Hence, if we use this information in (20), we get and from the hypothesis on , further we have Now, from Lemma 2, we obtain .

Theorem 8. Let be , positive real numbers, , complex number, , and . If , , then for any complex number , with , we have .

Proof. The proof is similar with Theorem 7 if we use Lemma 3, instead of Lemma 2.

Theorem 9. Let be , positive real numbers, , complex number, and complex number, , such that If , , then .

Proof. We consider again the function and after some calculations, we get In order to apply Lemma 4, we evaluate the expression and from (26), we get Since , according to Schwarz lemma, we obtain Further, since , if we put (28) in (27), we have But, since for , we have , from the last inequality, we get Further, if we use the condition from the hypothesis, related to , we obtain Now, from Lemma 4, we obtain .

Theorem 10. Let be , positive real numbers, , complex number, , and . If , , then for any complex number , with , we have .

Proof. We consider the function After some calculations, we obtain and and further Since , according to Schwarz lemma, we obtain Further, since , if we put (35) in (34), we have Using further the hypothesis and also , , from the last inequality, we get Now, if we apply Lemma 5 for , we obtain .

Remark 11. If in Theorems 7–10 we take different particular cases, as for example those from Remarks 1, we get the condition of univalence for different known integral operators. Theorems 7–9 extend the results obtained by Frasin in [8].

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflict of interest.

Acknowledgments

This research has been funded by the “1 Decembrie 1918” University of Alba Iulia, through the scientific research funds.