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Generalized conditions for starlikeness and convexity of certain analytic functions
Journal of Inequalities and Applications volume 2011, Article number: 87 (2011)
For analytic functions f(z) in the open unit disk with f (0) = 0 and f '(0) = 1, Nunokawa et al. (Turk J Math 34, 333-337, 2010)have shown some conditions for starlikeness and convexity of f(z). The object of the present paper is to derive some generalized conditions for starlikeness and convexity of functions f(z) with examples.
2010 Mathematics Subject Classification: Primary 30C45.
Let denote the class of functions f(z) of the form
which are analytic in the open unit disk . Let be the subclass of consisting of functions f(z) which are univalent in . A function is said to be starlike with respect to the origin in if is the starlike domain. We denote by the class of all starlike functions f(z) with respect to the origin in . Furthermore, if a function satisfies , then f(z) is said to be convex in . We also denote by the class of all convex functions in . Note that .
To discuss the univalency of , Nunokawa  has given
Lemma 1.1 If satisfies , then . Also, Mocanu  has shown that
Lemma 1.2 If satisfies
In view of Lemmas 1.1 and 1.2, Nunokawa et al.  have proved the following results.
Lemma 1.3 If satisfies
Lemma 1.4 If satisfies
The object of the present paper is to consider some generalized conditions for functions f(z) to be in the classes or .
2 Generalized conditions for starlikeness
We begin with the statement and the proof of generalized conditions for starlikeness.
Theorem 2.1 If satisfies
for some j(j = 2, 3, 4, ...), then , where
Proof For j = 2, the inequality (2.1) becomes (1.2) of Lemma 1.2. Thus, the theorem is hold true for j = 2. We need to prove the inequality for j ≧ 3. Note that
We suppose that Then, (2.3) gives us that
Therefore, if f(z) satisfies
then by Lemma 1.3. This means that if f(z) satisfies
then . Thus, the theorem is holds true for j = 3.
Next, we suppose that the theorem is true for j = 2, 3, 4, ..., (k - 1). Then, letting we have that
Thus, if f(z) satisfies
then . This is equivalent to
Therefore, the theorem holds true for j = k. Thus, applying the mathematical induction, we complete the proof of the theorem.
Example 2.1 Let us consider a function
if f(z) satisfies
then . This is equivalent to
Therefore, we put
Consequently, we see that the function
is in the class .
3 Generalized conditions for convexity
For the convexity of f(z), we derive
Theorem 3.1 If satisfies
for some j(j = 3, 4, 5, ...), then , where
Proof We have to prove for j ≧ 3. Note that
If then we have that
We know that if and only if . Therefore, if
then by means of Lemma 1.3. Thus, if
then . This shows that the theorem is true for j = 3.
Next, we assume that theorem is true for j = 3, 4, 5, ..., (k - 1). Then, letting we obtain that
Now, we consider . Then, (3.7) implies that
then (or ), if f(z) satisfies that
that is, that
then . Thus, the result is true for j = k. Using the mathematical induction, we complete the proof the theorem.
Example 3.1 We consider the function
Then, if f(z) satisfies
then . Since
With this conditions, the function
belongs to the class .
If we use the same technique as in the proof of Theorem 2.1 applying Lemma 1.4, then we have
Theorem 3.2 If satisfies
for some j (j = 2, 3, 4, ...), then , where M is given by (2.2).
Nunokawa M: On the order of strongly starlikeness of strongly convex functions. Proc Jpn Acad 1993, 68: 234–237.
Mocanu PT: Some starlikeness conditions for analytic function. Rev Roum Math Pures Appl 1988, 33: 117–124.
Nunokawa M, Owa S, Polatoglu Y, Caglar M, Duman EY: Some sufficient conditions for starlikeness and convexity. Turk J Math 2010, 34: 333–337.
This paper was completed when the first author was visiting Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan, from Atat ürk University, Turkey, between February 17 and 26, 2011.
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Uyanik, N., Owa, S. Generalized conditions for starlikeness and convexity of certain analytic functions. J Inequal Appl 2011, 87 (2011). https://doi.org/10.1186/1029-242X-2011-87
- Analytic function
- starlike function
- convex function