For any real number
${\it\beta}$
with
${\it\beta}>1$
, let
${\mathcal{M}}(\,{\it\beta})$
(
${\mathcal{N}}(\,{\it\beta})$
respectively) denote the class of analytic functions
$f$
in the unit disk
$\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$
of the form
$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$
and satisfying
$\text{Re}\,P_{f}<{\it\beta}$
(
$\text{Re}\,Q_{f}<{\it\beta}$
respectively) in
$\mathbb{D}$
, where
$P_{f}=zf^{\prime }(z)/f(z)$
and
$Q_{f}=1+zf^{\prime \prime }(z)/f^{\prime }(z)$
. Also, for
${\it\beta}>1$
, let
${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$
(
${\mathcal{N}}{\rm\Sigma}(\,{\it\beta})$
respectively) denote the class of analytic functions
$g$
of the form
$g(z)=z(1+\sum _{n=1}^{\infty }b_{n}z^{-n})$
and satisfying
$\text{Re}\,P_{g}<{\it\beta}$
(
$\text{Re}\,Q_{g}<{\it\beta}$
respectively) for
$z\in {\rm\Delta}=\{z\in \mathbb{C}:1<|z|<\infty \}$
. In this paper, we shall determine the coefficient bounds, inverse coefficient bounds, the growth and distortion theorem and the upper bounds for the Fekete–Szegő functional
${\rm\Lambda}_{{\it\lambda}}(f)=a_{3}-{\it\lambda}a_{2}^{2}$
for functions
$f$
in the classes
${\mathcal{M}}(\,{\it\beta})$
and
${\mathcal{N}}(\,{\it\beta})$
. Further, we shall solve the maximal area problem for functions of the type
$z/f(z)$
when
$f\in {\mathcal{M}}(\,{\it\beta})$
, which is Yamashita’s conjecture for the class
${\mathcal{M}}(\,{\it\beta})$
. We shall obtain the radius of convexity for the class
${\mathcal{N}}(\,{\it\beta})$
. We shall also determine the coefficient bounds for functions
$g$
in the classes
${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$
and
${\mathcal{N}}{\rm\Sigma}(\,{\it\beta})$
and the inverse coefficient bounds for functions
$g$
in the class
${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$
. All the results are sharp.