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On certain univalent functions with missing coefficients
Advances in Difference Equations volume 2013, Article number: 89 (2013)
Abstract
The main object of the present paper is to show certain sufficient conditions for univalency of analytic functions with missing coefficients.
MSC:30C45, 30C55.
1 Introduction
Let A(n) be the class of functions of the form
which are analytic in the unit disk U=\{z:z<1\}. We write A(2)=A.
A function f(z)\in A is said to be starlike in z<r (r\le 1) if and only if it satisfies
A function f(z)\in A is said to be closetoconvex in z<r (r\le 1) if and only if there is a starlike function g(z) such that
Let f(z) and g(z) be analytic in U. Then we say that f(z) is subordinate to g(z) in U, written f(z)\prec g(z), if there exists an analytic function w(z) in U, such that w(z)\le z and f(z)=g(w(z)) (z\in U). If g(z) is univalent in U, then the subordination f(z)\prec g(z) is equivalent to f(0)=g(0) and f(U)\subset g(U).
Recently, several authors showed some new criteria for univalency of analytic functions (see, e.g., [1–7]). In this note, we shall derive certain sufficient conditions for univalency of analytic functions with missing coefficients.
For our purpose, we shall need the following lemma.
Let f(z) and g(z) be analytic in U with f(0)=g(0). If h(z)=z{g}^{\prime}(z) is starlike in U and z{f}^{\prime}(z)\prec h(z), then
2 Main results
Our first theorem is given by the following.
Theorem 1 Let f(z)=z+{a}_{n}{z}^{n}+\cdots \in A(n) with f(z)\ne 0 for 0<z<1. If
where 0<\beta \le 2[1(n2){a}_{n}], then f(z) is univalent in U.
Proof
Let
then p(z) is analytic in U. By integration from 0 to z ntimes, we obtain
Thus, we have
where
It is easily seen from (2.1), (2.2) and (2.5) that
and, in consequence,
Since
we get
and so
for {z}_{1},{z}_{2}\in U and {z}_{1}\ne {z}_{2}.
Now it follows from (2.4) and (2.7) that
Hence, f(z) is univalent in U. The proof of the theorem is complete. □
Let {S}_{n}(\beta ) denote the class of functions f(z)=z+{a}_{n}{z}^{n}+\cdots \in A(n) with f(z)\ne 0 for 0<z<1, which satisfy the condition (2.1) given by Theorem 1.
Next we derive the following.
Theorem 2 Let f(z)=z+{a}_{n}{z}^{n}+\cdots \in {S}_{n}(\beta ). Then, for z\in U,
Proof In view of (2.1), we have
Applying Lemma to (2.11), we get
By using the lemma repeatedly, we finally have
According to a result of Hallenbeck and Ruscheweyh [[1], Theorem 1], (2.13) gives
i.e.,
where w(z) is analytic in U and w(z)\le {z}^{n1} (z\in U).
Now, from (2.15), we can easily derive the inequalities (2.8), (2.9) and (2.10). □
Finally, we discuss the following theorem.
Theorem 3 Let f(z)\in {S}_{n}(\beta ) and have the form

(i)
If \frac{2}{\sqrt{5}}\le \beta \le 2, then f(z) is starlike in z<\sqrt[n]{\frac{2}{\beta}}\cdot \frac{1}{\sqrt[2n]{5}};

(ii)
If \sqrt{3}1\le \beta \le 2, then f(z) is closetoconvex in z<\sqrt[n]{\frac{\sqrt{3}1}{\beta}}.
Proof
If we put
then by (2.1) and the proof of Theorem 2 with {a}_{n}=0, we have
It follows from the lemma that
which implies that

(i)
Let \frac{2}{\sqrt{5}}\le \beta \le 2 and
z<{r}_{1}=\sqrt[n]{\frac{2}{\beta}}\cdot \frac{1}{\sqrt[2n]{5}}.(2.21)
Then by (2.20), we have
Also, from (2.8) in Theorem 2 with {a}_{n}=0, we obtain
and so
Therefore, it follows from (2.22) and (2.24) that
for z<{r}_{1}. This proves that f(z) is starlike in z<{r}_{1}.

(ii)
Let \sqrt{3}1\le \beta \le 2 and
z<{r}_{2}=\sqrt[n]{\frac{\sqrt{3}1}{\beta}}.(2.25)
Then we have
Thus, Re{f}^{\prime}(z)>0 for z<{r}_{2}. This shows that f(z) is closetoconvex in z<{r}_{2}. □
References
Dziok J, Srivastava HM: Certain subclasses of analytic functions associated with the generalized hypergeometric function. Integral Transforms Spec. Funct. 2003, 14: 7–18. 10.1080/10652460304543
Nunokawa M, Obradovič M, Owa S: One criterion for univalency. Proc. Am. Math. Soc. 1989, 106: 1035–1037. 10.1090/S00029939198909756535
Obradovič M, Pascu NN, Radomir I: A class of univalent functions. Math. Jpn. 1996, 44: 565–568.
Owa S: Some sufficient conditions for univalency. Chin. J. Math. 1992, 20: 23–29.
Samaris S: Two criteria for univalency. Int. J. Math. Math. Sci. 1996, 19: 409–410. 10.1155/S0161171296000579
Silverman H: Univalence for convolutions. Int. J. Math. Math. Sci. 1996, 19: 201–204. 10.1155/S0161171296000294
Yang DG, Liu JL: On a class of univalent functions. Int. J. Math. Math. Sci. 1999, 22: 605–610. 10.1155/S0161171299226051
Hallenbeck DJ, Ruscheweyh S: Subordination by convex functions. Proc. Am. Math. Soc. 1975, 51: 191–195. 10.1090/S0002993919750402713X
Suffridge TJ: Some remarks on convex maps of the unit disk. Duke Math. J. 1970, 37: 775–777. 10.1215/S0012709470037920
Acknowledgements
Dedicated to Professor Hari M Srivastava.
We would like to express sincere thanks to the referees for careful reading and suggestions, which helped us to improve the paper.
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Cang, YL., Liu, JL. On certain univalent functions with missing coefficients. Adv Differ Equ 2013, 89 (2013). https://doi.org/10.1186/16871847201389
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DOI: https://doi.org/10.1186/16871847201389