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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
 1.
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
 2.
Nunokawa M, Obradovič M, Owa S: One criterion for univalency. Proc. Am. Math. Soc. 1989, 106: 1035–1037. 10.1090/S00029939198909756535
 3.
Obradovič M, Pascu NN, Radomir I: A class of univalent functions. Math. Jpn. 1996, 44: 565–568.
 4.
Owa S: Some sufficient conditions for univalency. Chin. J. Math. 1992, 20: 23–29.
 5.
Samaris S: Two criteria for univalency. Int. J. Math. Math. Sci. 1996, 19: 409–410. 10.1155/S0161171296000579
 6.
Silverman H: Univalence for convolutions. Int. J. Math. Math. Sci. 1996, 19: 201–204. 10.1155/S0161171296000294
 7.
Yang DG, Liu JL: On a class of univalent functions. Int. J. Math. Math. Sci. 1999, 22: 605–610. 10.1155/S0161171299226051
 8.
Hallenbeck DJ, Ruscheweyh S: Subordination by convex functions. Proc. Am. Math. Soc. 1975, 51: 191–195. 10.1090/S0002993919750402713X
 9.
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, Y., Liu, J. On certain univalent functions with missing coefficients. Adv Differ Equ 2013, 89 (2013). https://doi.org/10.1186/16871847201389
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Keywords
 analytic
 univalent
 subordination