  For $0<\lambda\le 1$, let $\mathcal{U}(\lambda)$ be the class analytic
functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}$
satisfying $|f'(z)(z/f(z))^2-1|<\lambda$ and $\mathcal{U}:=\mathcal{U}(1)$. In
the present article, we prove that the class $\mathcal{U}$ is contained in the
closed convex hull of the class of starlike functions and using this fact, we
solve some extremal problems such as integral mean problem and arc length
problem for functions in $\mathcal{U}$. By means of the so-called theory of
star functions, we also solve the integral mean problem for functions in
$\mathcal{U}(\lambda)$. We also obtain the estimate of the Fekete-Szeg\"{o}
functional and the pre-Schwarzian norm of certain nonlinear integral transform
of functions in $\mathcal{U}(\lambda)$. Further, for the class of meromorphic
functions which are defined in
$\Delta:=\{\zeta\in\mathbb{\widehat{C}}:|\zeta|>1\}$ and associated with the
class $\mathcal{U}(\lambda)$, we obtain a sufficient condition for a function
$g$ to be an extreme point of this class.
