1. Introduction
Let
denote the class of functions
that are analytic in the unit
disk Let be
the subclass of consisting of
functions of the form which are univalent
in . Furthermore, let be the subclass of consisting of functions of the form
The class of
starlike functions of order
is defined
as In particular, is the class of starlike
functions with respect to the origin.
The class of convex
functions of order is defined by
or equivalently, as introduced by
Robertson [1]. Note that
, where represents the class of functions
such that is starlike with respect to the
origin. Similarly, , the
well-known class of convex functions. It is a well-established fact that
if and only if
.
In [2], Opoola introduced
the following differential operator ,
defined as: and, for , If is
given by (1), then from (3) we have
where , , , and .
When and , was
introduced by Salagean [3].
When and , was defined by Al-Oboudi [4].
The study of geometric properties of analytic functions plays a vital
role in complex analysis. Researchers have extensively investigated
properties such as radii of starlikeness, convexity, close-to-convexity,
and growth and distortion theorems.
Gupta and Jain [5]
introduced the subclasses and for and
established geometric properties such as coefficient inequalities,
growth and distortion theorems, closure under arithmetic mean and linear
combinations, integral representations, and extreme point theorems.
Kulkarni [6] further extended
this by introducing the subclass , proving similar geometric properties and deriving
integral representations.
Sambo and Opoola [7]
studied a subclass of analytic functions, obtaining characterization
properties such as radii of starlikeness, convexity, close-to-convexity,
and growth and distortion results.
Motivated by these works and contributions from [8-26],
we define new subclasses of analytic functions and derive their
geometric properties, including characterization properties, growth and
distortion theorems, arithmetic means, and radii of convexity. Our
results generalize several existing findings. We introduce new
subclasses of starlike and convex functions as follows:
Definition 1. A function of the form (1) is said to
belong to the class
if it satisfies the following condition: where , , , , , and is the Opoola
differential operator defined in (4).
Definition 2. A function of the form (1) is said to
belong to the class
if it satisfies the following condition: where , , , , , and is the Opoola
differential operator defined in (4).
Let
and
denote subclasses of in (2), defined as:
2. Main Results
This section presents the main results of this study.
2.1. Characterization
Properties of the Classes:
Theorem 3. A function of the form (1) belongs to the class
if where The result in (7) is sharp
for functions of the form
Proof. Suppose that (7) holds. For
, we have By the maximum modulus theorem, which implies
Therefore, This completes the
proof. 
Theorem 4. If a function of the form (2) is in the
class
then
The result in (11) is sharp
for functions of the form
Proof. It suffices to prove and only if part. If in (2) belongs to
the class ,
then
We know that , so that Taking values of on the real line and making , we have Which is the required result. 
Theorem 5. A function of the form (1) is in the class
if
The result in (16) is sharp
for functions of the form
Proof. Suppose (16) holds. Taking , and using the inequality , we have: By the Maximum Modulus Theorem, we obtain: Hence, we have: Therefore, . The proof is
complete. 
Theorem 6. If a function of the form (2) is in the
class
then The result in (20) is sharp
for functions of the form
Proof. It suffices to prove the if part only. Assume in (2) belongs to
the class ,
then
We know that , so Taking values of on the real line and making , we have Which is the required result. 
2.2. Growth and Distortion
Theorem
Theorem 7. If , then for ,
Proof. Let . By Theorem 4, we have and Hence, and Therefore, and Also, and Thus, and Hence, the proof is
complete. 
Theorem 8. If , then for ,
Proof. Let . By Theorem 6, we have and Hence, we have the
following inequalities for :
Therefore, and For the derivative
, we have
Thus, and Hence, the proof
is complete. 
2.3. Arithmetic Mean
We assert that the classes
and
are closed under arithmetic mean.
Theorem 9. If
and
are in the class
then,
Proof. Since and
, from Theorem 4, we
have Therefore, we obtain Hence, , and the proof is
complete. 
Theorem 10. If
and
are in the class
then,
Proof. The proof holds same as Theorem 9. 
2.4. Radius of convexity
Theorem 11. If , then is convex of order in , where
Proof. Let . Then, we have the
inequality Now,
consider Taking the modulus, we get
Rearranging, we obtain Simplifying further: From Theorem 3, we know that
This inequality holds if
Thus, which completes the
proof. 
3. Conclusion
It is clear that the new classes studied in this work generalise some
well-known classes of analytic and univalent functions. Also, the
results in this study generally reduce to some well-known and new
results with appropriate variations of the involved parameters. The new
classes however, apparently generalised many existing ones and the
results from this research extended many known and new ones when the
underlying parameters are varied. Invariably, these results augment
those that are already existing in literature.