- Open Access
Coefficient, distortion and growth inequalities for certain close-to-convex functions
© Cho et al; licensee Springer. 2011
- Received: 24 June 2011
- Accepted: 27 October 2011
- Published: 27 October 2011
In the present investigation, certain subclasses of close-to-convex functions are investigated. In particular, we obtain an estimate for the Fekete-Szegö functional for functions belonging to the class, distortion, growth estimates and covering theorems.
Mathematics Subject Classification (2010): 30C45, 30C80.
- starlike functions
- close-to-convex functions
- Fekete-Szegö inequalities
- distortion and growth theorems
- subordination theorem
for some function g ∈ S*(1/2). The idea here is to replace the average of f(z) and - f(-z) by the corresponding product -g(z) g(-z), and the factor z is included to normalize the expression, so that -z2f'(z)/(g(z) g(-z)) takes the value 1 at z = 0. To make the functions univalent, it is further assumed that g is starlike of order 1/2 so that the function -g(z) g(-z)/z is starlike, which in turn implies the close-to-convexity of f. For some recent works on the problem, see [4–7]. Instead of requiring the quantity -z2f'(z)/(g(z) g(-z)) to lie in the right-half plane, we can consider more general regions. This could be done via subordination between analytic functions.
Let f and g be analytic in . Then f is subordinate to g, written f ≺ g or , if there is an analytic function w(z), with w(0) = 0 and |w(z)| < 1, such that f(z) = g(w(z)). In particular, if g is univalent in , then f is subordinate to g, if f(0) = g(0) and . In terms of subordination, a general class is introduced in the following definition.
for some function g ∈ S*(1/2).
This class was introduced by Wang et al. . A special subclass where φ(z): = (1 + (1 - 2γ) z)/(1 - z), 0 ≤ γ < 1, was recently investigated by Kowalczyk and Leś-Bomba . They proved the sharp distortion and growth estimates for functions in as well as some sufficient conditions in terms of the coefficient for function to be in this class .
In the present investigation, we obtain a sharp estimate for the Fekete-Szegö functional for functions belonging to the class . In addition, we also investigate the corresponding problem for the inverse functions for functions belonging to the class . Also distortion, growth estimates as well as covering theorem are derived. Some connection with earlier works is also indicated.
In this section, we assume that the function φ(z) is an univalent analytic function with positive real part that maps the unit disk onto a starlike region which is symmetric with respect to real axis and is normalized by φ'(0) = 1 and φ(0) > 0. In such case, the function φ has an expansion of the form φ(z) = 1 + B1z + B2z2 +..., B1 > 0.
for an analytic function w with w(0) = 0 and |w(z)| < 1 which is sharp for the functions w(z) = z2 or w(z) = z, the desired result follows upon using the estimate that for analytic function g(z) = z + g2z2 + g3z3 +... which is starlike of order 1/2.
The functions f0 and f1 show that the results are sharp.
while the limiting case μ → ∞ gives the sharp estimate |a2| ≤ B1/2. In the special case where φ(z) = (1 + z)/(1 - z), the results reduce to the corresponding one in [3, Theorem 2, p. 125].
Though Xu et al.  have given an estimate of |a n | for all n, their result is not sharp in general. For n = 2, 3, our results provide sharp bounds.
Our result follows at once from this identity and Theorem 1.
The second coefficient of univalent function plays an important role in the theory of univalent function; for example, this leads to the distortion and growth estimates for univalent functions as well as the rotation theorem. In the next theorem, we derive the distortion and growth estimates for the functions in the class . In particular, if we let r → 1- in the growth estimate, it gives the bound |a2| ≤ B1/2 for the second coefficient of functions in .
The other inequality for |f'(z)| is similar. Since the function f is univalent, the inequality for |f(z)| follows from the corresponding inequalities for |f'(z)| by Privalov's Theorem [10, Theorem 7, p. 67].
Thus, the function f0 satisfies the subordination (1) with g0, while the function f1 satisfies it with g1; therefore, these functions belong to the class . It is clear that the upper estimates for |f'(z)| and |f(z)| are sharp for the function f0 given in (6), while the lower estimates are sharp for fl given in (6).
Remark 2 We note that Xu et al.  also obtained a similar estimates and our results differ from their in the hypothesis. Also we have shown that the results are sharp. Our hypothesis is same as the one assumed by Ma and Minda .
where |z| = r < 1. Also our result improves the corresponding results in .
Remark 4 Let . Then the disk for every .
It is well known  that f is starlike if (1 - t) f(z) ≺ f(z) for t ∈ (0, ∈), where ∈ is a positive real number; also the function is starlike with respect to symmetric points if (1 - t) f(z) + tf(-z) ≺ f(z). In the following theorem, we extend these results to the class . The proof of our result is based on the following version of a lemma of Stankiewicz .
Theorem 3 Letand(1/2). Let ∈ > 0 and f(z) + tg(z)g(-z)/z ≺ f(z), t ∈ (0, ∈). Then.
The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2011-0007037).
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