U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A
VOL. LXXIII, NO. 1, 2019 SECTIO A 27–31
SZYMON IGNACIUK and MACIEJ PAROL
Criteria of univalence for a certain integral operator
Abstract. In this article we consider the problem of univalence of a function introduced by Breaz and Ularu, improve some of their results and receive not only univalence conditions but also close-to-convex conditions for this function. To this aim, we used our method based on Kaplan classes.
1. Introduction. We consider the following subclasses of the class of all analytic functions in the unit disk D := {z ∈ C : |z| < 1}:
• A as the class of all functions f normalized by f (0) = f0(0) − 1 = 0,
• H as the subclass of A of all functions f that are locally univalent, i.e., f0 6= 0 in D,
• S as the class of all univalent functions belonging to A,
• K as the class of functions in S that map D onto a convex set,
• C as the class of functions in S that are close-to-convex,
• Hd as the class of all analytic functions f normalized by f (0) = 1 and such that f 6= 0 in D.
Univalence of integral operators for the functions from known classes K, C and S was studied by many authors (see [1, 3, 4, 6–8]). In this article we consider univalence of slightly modified integral operator introduced in [2].
2010 Mathematics Subject Classification. 30C45, 30C55, 44A05.
Key words and phrases. Univalence, integral operators, Kaplan classes.
For α, β ≥ 0, the Kaplan class K(α, β) is a set of all functions f ∈ Hd satisfying the condition
(1.1) −απ −1
2(α − β)(θ1− θ2) ≤ arg f (reiθ2) − arg f (reiθ1) for 0 < r < 1 and θ1< θ2 < θ1+ 2π (see [9, pp. 32–33]).
Let us recall [9, p. 46] that:
• f ∈ K if and only if f0 ∈ K(0, 2),
• f ∈ C if and only if f0∈ K(1, 3).
First we will call the following lemmas from [5].
Lemma A. For all α1, α2, β1, β2≥ 0 and t ∈ R\{0} the following conditions hold:
f ∈ K(α1, β1) and g ∈ K(α2, β2) ⇒ f g ∈ K(α1+ α2, β1+ β2) , f ∈ K(α1, β1) ⇐⇒ ft∈ K |t| + t
2 α1+|t| − t
2 β1,|t| + t
2 β1+|t| − t 2 α1
, f ∈ K(α1, β1) ⇒ f0∈ K(0, 0) .
Lemma B. For all α1, α2, β1, β2 ≥ 0 the following equivalences hold:
α1 ≤ α2 ⇐⇒ K(α1, β1) ⊂ K(α2, β1) , β1 ≤ β2 ⇐⇒ K(α1, β1) ⊂ K(α1, β2) .
Now we will define an integral operator which is the subject of our re- search.
Definition 1.1. Let ν ∈ C. For all functions f, g ∈ H we define the function (1.2) D 3 z 7→ F (z; f, g; ν) :=
z
Z
0
(f0(u))|ν|eνg(u)du .
2. Main results. In this section we will show our criteria of univalence of the integral operator given by (1.2) and we compare them with results from [2].
Theorem 2.1. Let f ∈ K, g ∈ H and |g(z)| ≤ M for all z ∈ D and a certain M ≥ 0. If
(2.1) |ν| ≤
( 3π
2(M +π), for M < π2 ,
π
2M , for M ≥ π2 ,
then F (·; f, g; ν) ∈ C.
Proof. Fix f ∈ K, g ∈ H and |g(z)| ≤ M for all z ∈ D and a certain M ≥ 0.
We know that f0 ∈ K(0, 2) and by Lemma A we obtain (f0)|ν|∈ K(0, 2|ν|).
On the other hand, we get (2.2)
arg
eνg(z)
= | Im(νg(z))| ≤ |νg(z)| ≤ |ν|M
for z ∈ D. Consider (1.1) with f := eνg. Then for D 3 z := reiθ and 0 ≤ α ≤ β we get
arg
eνg(reiθ2)
− arg
eνg(reiθ1)
≥ −2|ν|M ≥ −απ − 1
2(α − β) · 0 = −απ and for 0 ≤ β < α we get
arg
eνg(reiθ2)
−arg
eνg(reiθ1)
≥ −2|ν|M ≥ −απ−1
2(α−β)·(−2π) = −βπ.
As a consequence eνg ∈ K(α, β) for α, β ≥ 2|ν|M/π. This and Lemma A show that
F0(·; f, g; ν) ∈ K 2|ν|M
π ,2|ν|(M + π) π
. From Lemma B we know that F0(·; f, g; ν) ∈ C if
2|ν|M π ≤ 1 and
2|ν|(M + π)
π ≤ 3.
Therefore, F0(·; f, g; ν) ∈ C if
|ν| ≤
( 3π
2(M +π), for M < π2,
π
2M, for M ≥ π2.
Remark 2.2. Let us notice that the assumptions for functions f and g in Theorem 2.1 are much weaker than assumptions in [2, Theorem 2.1]. Since for any z ∈ D we have
f00(z) f0(z)
≤ 1 =⇒
zf00(z) f0(z)
≤ |z| < 1 =⇒ Re
1 +zf00(z) f0(z)
> 0 =⇒ f ∈ K, so
f ∈ H :
f00(z) f0(z)
≤ 1 for every z ∈ D
⊂ K.
The inclusion can not be replaced by an equality. For example, the function D 3 z 7→ f (z) := z/(1 − z) satisfies the condition
f ∈ K \
f ∈ H :
f00(z) f0(z)
≤ 1 for every z ∈ D
.
Moreover, we assume that g is only bounded while in [2] there is an addi- tional condition for g. For ν ∈ C \ R results are incomparable, since we use |ν| in the definition of F (·; f, g; ν). However, in this article there is no
additional restriction that Re(ν) ≥ 0. For ν > 0 results from Theorem 2.1 can be directly compared with [2, Theorem 2.1]. For
M >
√3 + q
3 + 8π2(√ 3 − 1)
4π ≈ 0.758
results obtained in Theorem 2.1 are better with much weaker assumptions.
Moreover, let us point out that in this article we prove that F (·; f, g; ν) ∈ C and not only that F (·; f, g; ν) ∈ S.
The following theorem has weaker assumptions about a function f than Theorem 2.1 but still improves results from [2] in some cases.
Theorem 2.3. Let f ∈ C, g ∈ H and |g(z)| ≤ M for all z ∈ D and a certain M ≥ 0. If
(2.3) |ν| ≤ π
2M + π , then F (·; f, g; ν) ∈ C.
Proof. Fix f ∈ C, g ∈ H and |g(z)| ≤ M for all z ∈ D and a certain M ≥ 0.
We know that f0 ∈ K(1, 3) and by Lemma A we obtain (f0)|ν|∈ K(|ν|, 3|ν|).
Analogously to the proof of Theorem 2.1 we get F0(·; f, g; ν) ∈ K |ν|(2M + π)
π ,|ν|(2M + 3π) π
. From Lemma B we know that F0(·; f, g; ν) ∈ C if
|ν|(2M + π)
π ≤ 1
and |ν|(2M + 3π)
π ≤ 3 .
Therefore, F0(·; f, g; ν) ∈ C if
|ν| ≤ π 2M + π .
Remark 2.4. Let us notice that in Theorem 2.3 we weaken the assumptions for f with respect to Theorem 2.1. Analogously as in Remark 2.2 for ν > 0 results from Theorem 2.3 can be directly compared with [2, Theorem 2.1].
For
M > 6√ 3 +
q
108 + 16π2(3√ 3 − 2)
8π ≈ 1.398
the results obtained in Theorem 2.3 are better with much weaker assump- tions for f , g and ν. Moreover, let us point out that in this article we prove that F (·; f, g; ν) ∈ C and not only that F (·; f, g; ν) ∈ S.
References
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Math. Letters 25 (3) (2012), 658–661.
[3] Causey, W. M., The univalence of an integral, Proc. Amer. Math. Soc. 3 (1971), 500–
502.
[4] Godula, J., On univalence of a certain integral, Ann. Univ. Mariae Curie-Skłodowska Sect. A 33 (1979), 69–76.
[5] Ignaciuk, S., Parol, M., Kaplan classes and their applications in determining univa- lence of certain integral operators, 2018 (to appear).
[6] Krzyż, J., Lewandowski, Z., On the integral of univalent functions, Bull. Pol. Acad.
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Math. Soc. 27 (1971), 97–100.
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[9] Ruscheweyh, S., Convolutions in Geometric Function Theory, Seminaire de Math.
Sup. 83, Presses de l’Universit´e de Montr´eal, 1982.
Szymon Ignaciuk Maciej Parol
Department of Applied Mathematics Institute of Mathematics and Computer Sciences and Computer Science
University of Life Sciences The John Paul II Catholic University of Lublin
ul. Akademicka 13 ul. Konstantynów 1H
20-950 Lublin 20-708 Lublin
Poland Poland
e-mail: szymon.ignaciuk@gmail.com
Institute of Mathematics
Maria Curie-Skłodowska University pl. Marii Curie-Skłodowskiej 1 20-031 Lublin
Poland
e-mail: Maciej.Parol@Live.umcs.edu.pl Received September 3, 2018
