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This article is cited in 10 scientific papers (total in 10 papers)
Coefficient problems on the class $U(\lambda)$
Saminathan Ponnusamya, Karl-Joachim Wirthsb a Department of Mathematics,
Indian Institute of Technology Madras,
Chennai-600 036, India
b Institut für Analysis und Algebra,
TU Braunschweig,
38106 Braunschweig, Germany
Abstract:
For $0<\lambda \leq 1$, let ${\mathcal U}(\lambda)$ denote the family of
functions $f(z)=z+\sum\limits_{n=2}^{\infty}a_nz^n$
analytic in the unit disk $\mathbb{D}$ satisfying the condition $\left |\left (\frac{z}{f(z)}\right )^{2}f'(z)-1\right |<\lambda $
in $\mathbb{D}$. Although functions in this family are known to be univalent in $\mathbb{D}$, the coefficient conjecture about $a_n$
for $n\geq 5$ remains an open problem. In this article, we shall
first present a non-sharp bound for $|a_n|$. Some members of the family ${\mathcal U}(\lambda)$ are given by
$$ \frac{z}{f(z)}=1-(1+\lambda)\phi(z) + \lambda (\phi(z))^2
$$
with $\phi(z)=e^{i\theta}z$, that solve many extremal problems
in ${\mathcal U}(\lambda)$. Secondly, we shall consider the following question: Do there exist functions
$\phi$ analytic in $\mathbb{D}$ with $|\phi (z)|<1$ that are not of the form $\phi(z)=e^{i\theta}z$
for which the corresponding functions $f$ of the above form are members of the family ${\mathcal U}(\lambda)$?
Finally, we shall solve the second coefficient ($a_2$) problem in an explicit form for $f\in {\mathcal U}(\lambda)$
of the form
$$f(z) =\frac{z}{1-a_2z+\lambda z\int\limits_0^z\omega(t)\,dt},
$$
where $\omega$ is analytic in $\mathbb{D}$ such that $|\omega(z)|\leq 1$ and $\omega(0)=a$, where $a\in \overline{\mathbb{D}}$.
Keywords:
Univalent function; subordination; Julia's lemma; Schwarz' lemma.
Received: 26.12.2017 Revised: 10.03.2018 Accepted: 12.03.2018
Citation:
Saminathan Ponnusamy, Karl-Joachim Wirths, “Coefficient problems on the class $U(\lambda)$”, Probl. Anal. Issues Anal., 7(25):1 (2018), 87–103
Linking options:
https://www.mathnet.ru/eng/pa226 https://www.mathnet.ru/eng/pa/v25/i1/p87
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