Abstract
In this paper, we determine conditions on \(\beta , \alpha _{i}\) and \(g_{i}\) such that the integral operator \({ G_{\alpha _1, \alpha _2,\ldots ,\alpha _n, \beta }}\) is univalent on the exterior of the unit disk for two subclasses of analytical functions.
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1 Introduction
Let O be the class of analytical functions g defined on the exterior of the unit disk \(W=\{ z \in \mathbb {C} : 1< |z|<\infty \}\).
Let \(\sum \) be the subclass of O which contains the univalent functions of W.
Let \(O_1\) be the subclass of O which contains the meromorphic, normalized and injective functions \(g : W\longrightarrow \mathbb {C}_{\infty }, \) that looks like [2]:
With \(g(\infty )= \infty , g'(\infty )=1.\)
Let V be the subclass of univalent functions from O such that:
Let \(V_2 \) be the subclass of V. Let \(V_{2,\mu }\) be the subclass of \(V_2 \) which contains the functions of the form (1.1) and satisfies the condition:
for \(\mu >1 \) and we note \(V_{2,1} = V_2. \)
Let \(p\in \mathbb {R}\), with \(1< p < 2,\) S(p) is the subclass of O with all the functions such that:
Then we obtain the inequation:
Let A be the class of analytic functions f(z) defined in the open unit disk \(U:= \{z\in \mathbb {C}: |z|< 1 \}\) and normalized by the conditions:
Let S be the subclass of A consisting of univalent functions in U, of the form:
Let T be the univalent subclass of A which satisfies:
Let \(T_2\) be the subclass of T for which \( f''(0)=0.\)
It is known that between the S class and the \(\sum \) class there are the following links:
Proposition 1.1
[2]
-
(i)
Let f \(\in \)S and g(\(\varsigma \))=1/f(1/\(\varsigma \)), \(\varsigma \in W\). Then g \(\in \sum \) and g\((\varsigma )\ne \)0, \(\varsigma \in W\).
-
(ii)
If g \(\in \sum \) and g\((\varsigma )\ne \)0, \(\varsigma \in W\), then f \(\in \)S where f(z)=1/g(1/z), \(z \in U\).
Let \(F_{\alpha _1, \alpha _2,\ldots ,\alpha _n, \beta }\) be the integral operator introduced by Daniel Breaz and Narayanasamy Seenivasagan [3]:
and we take into account that \(f_i (t) \in S.\)
(\(f_i (t) \in T_2 \) which is a subclass of T, which is a subclass of A).
Let be \(g_i(t) = \frac{1}{f_i \left( \frac{1}{t}\right) } \in O_1\), with \(g_i(t)\ne 0; t \in O_1\)
We remember that \(O_1\) is the subclass of O with:
We may say that between \(T_2 \) and \(O_1 \) there is a bijection.
We start from:
And we apply the following transformations:
We can form the integral operator:
So:
Pascu proved the following theorem:
Theorem 1.1
[4, 5] Let \(\beta \in \mathbb {C}, Re \beta \ge \gamma > 0\). If the function \(f \in A\) satisfies the condition:
then the integral operator:
Theorem 1.2
[6] Let \(\alpha , \beta \in \mathbb {C}\) and \(Re \beta \ge Re \alpha \ge \frac{3}{|\alpha |}\). If the function \(f \in T_2\) satisfies the condition:
then the integral operator:
is in S.
Using Theorem 1.1 and Theorem 1.2, D. Breaz and N. Breaz obtained the following theorem:
Theorem 1.3
[1] Let \(\alpha , \beta \in \mathbb {C} \) and \(Re \beta \ge Re \alpha \ge \frac{3n}{|\alpha |}, \) let \(f_i \in T_2\) with:
and if \( |f_i (z)|\le 1, z\in U,\)
then the integral operator:
is in S.
2 Main results
Applying the Theorem 1.1 for the transformation \(t \rightarrow \frac{1}{t} | ( ) ' , \) we obtain:
Theorem 2.1
Let \(\beta \in \mathbb {C}, Re \beta \ge \gamma > 0\). If \( f \in T_2, g\in V_2\) satisfies the conditions:
and
then the integral operator:
is in \(\sum .\)
Applying the Theorem 1.2 for the transformation \(t \rightarrow \frac{1}{t} | ( ) ' , \) we get :
Theorem 2.2
Let \(\alpha , \beta \in \mathbb {C}\) and \(Re \beta \ge Re \alpha \ge \frac{3}{|\alpha |}\). If \(f\in T_2, g \in V_2\) satisfies the condition:
and \(\bigg | f(\frac{1}{z})\bigg |=\bigg | \frac{1}{g(z)}\bigg | \ge 1, z \in W\), then the integral operator:
is in W.
Using Theorem 1.3, Theorem 2.1 and Theorem 2.2 for the transformation \(t \rightarrow \frac{1}{t} | ( ) ', \) we get the following theorem:
Theorem 2.3
Let \(\alpha , \beta \in \mathbb {C} \) and \(Re \beta \ge Re \alpha \ge \frac{3n}{|\alpha |}\)
Let \(g_i \in V_2\), with:
and if \( |g_i (z)|>1, z\in W, \)
then the integral operator:
is in \(\sum .\)
Theorem 2.4
Let \(m > 1, g_i \in V_{2,\mu _i}\) defined by (2.5), \(\alpha _i, \beta \in \mathbb {C}, Re \beta \ge \gamma \) and:
If \(\Bigg | g_i(z) \Bigg | > m, z \in W, i = 1, 2, \ldots , n,\)
then we obtain that the integral operator from (1.5) \( G_{\alpha _i, \beta } \) is in \(\sum .\)
Proof
Let be the function:
We notice that:
We know that: \( \Bigg | g_i(z) \Bigg | > m\)
z \(\in W, i=1, 2, \ldots , n\) \(\square \)
Applying the relation (2.9) in the relation (2.8) we get:
We know that:
Using the results from the Eqs. (2.10) and (2.11) we get:
Given that \(Re \beta \ge \gamma > 0\) result from Theorem 1.1. that:
is in \(\sum .\)
Meaning that the integral operator:
is in \(\sum \).
Theorem 2.5
Let \(m > 1, g_i \in S(p)\) and:
If
Then we obtain that the integral operator defined in (1.5) \( G_{\alpha _i, \beta } \) is in \(\sum .\)
The proof of this theorem is very similar with the previous one.
References
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Kohr, G., Mocanu, P.: Capitole speciale de analiză complexă, pp. 133–144 (2020)
Seenivagan, N., Breaz, D.: Certain sufficient conditions for univalence. General Math. 15(4), 7–15 (2007)
Pascu, N.N.: An improvement of Becker’s univalence criterion. In: Proceedings of the commemorative session: Simion Sto\(\ddot{\iota }\)low (Braşov, 1987), pp. 43–48, Univ. Braşov, Braşov (1987)
Pascu, N.N.: On a univalence criterion. II. In: Itinerant seminar on functional equations, approximation and convexity (Cluj-Napoca, 1985), pp. 153–154, Univ. Babeş-Bolyai”, Cluj (2020)
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Pârva, O.M., Breaz, D.V. Univalence properties of an integral operator. Afr. Mat. 33, 37 (2022). https://doi.org/10.1007/s13370-022-00975-0
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DOI: https://doi.org/10.1007/s13370-022-00975-0