On a geometric study of a class of normalized functions defined by Bernoulli’s formula

The central purpose of this effort is to investigate analytic and geometric properties of a class of normalized analytic functions in the open unit disk involving Bernoulli’s formula. As a consequence, some solutions are indicated by the well-known hypergeometric function. The class of starlike functions is investigated containing the suggested class.


Introduction
Ma and Minda offered a couple classes of starlike and convex normalized functions in the open unit disk. They defined these classes utilizing the theory of differential subordination. Later, the Ma-Minda classes were considered by many investigators. Furthermore, the researchers established different classes containing different types of linear, differential, and integral operators [1]. Denote the class of analytic functions in the open unit disk by H(∪), and for ∈ C, we define the class H[ς, k] = ψ ∈ H(∪) : ψ(ξ ) = ς + ς k ξ k + ς k+1 ξ k+1 + · · · . Define a subclass of H[0, 1] as follows: For a normalized analytic function σ (ξ ) ∈ ∧, ξ ∈ ∪ := {ξ ∈ C : |ξ | < 1} satisfying the power series σ (ξ ) = ξ + ∞ n=2 a n ξ n , ξ ∈ ∪, (1.1) the Ma-Minda starlike class (M * ) is defined as follows: 1), (1.2) where ς is an analytic function with positive real part on , ς(0) = 1, ς (0) > 0, and ς maps onto a starlike domain corresponding to ∂ and symmetric with respect to the real axis. The symbol ≺ is presented as the notion of the subordination (see [2]). Moreover, the Ma-Minda convex class (M c ) is formulated by the subordination inequality A linear combination of these two classes is formulated by using different types of analytic functions including differential, integral, and linear operators and transformations (see for recent studies [3][4][5][6][7][8][9][10][11][12]). In this note, we suggest to study the general formula of Bernoulli's equation in a complex domain. We formulate an integral operator and a transform convoluted with a special function concerning Bernoulli's formula subordinated with a class of analytic functions. We present sufficient conditions to be a starlike function. Special cases are illustrated in the sequel.

Bernoulli's formula
In this section, we present a special type of Bernoulli's equation as follows: (2.1) The solution of (2.1) is given by A first generalization of (2.1) is formulated by considering a convex formula where α ∈ [0, 1] taking the solution formula It is clear that when α = 0.5, Eq. (2.2) reduces to Eq. (2.1). The most parametric Bernoulli's equation is given by having the solution When β ∈ [0, α), Bernoulli's formula is studied early [13,14]. In this effort, we consider β ∈ [1, ∞) and hence, for β = 2, we have two solutions as follows: and for β = 3, we get three different solutions More generalization can be viewed by consider the following equation: For example, let λ(ξ ) = ξ (starlike in ∪), then Eq. (2.4) admits a unique solution satisfying the equation Also, let λ(ξ ) = e ξ -1, which is starlike in ∪ (see [2], p.270), then the solution becomes where indicates the incomplete gamma function. We proceed to suggest a class of analytic functions taking the Ma-Minda design as follows.
The aim of the above class is to dominate the normalized solution of Bernoulli's equation by another normalized function in ∧.
Proof For the first part of the theorem, we suppose that Then, by the Carathéodory positivist method for analytic functions, we have where dμ is a probability measure. For the second part, we have the assumption Then, according to [15]-Theorem 1.6(P22) and for some real numbers , we obtain The next theorem is the converse of Theorem 3.1.
We proceed to present more information about solutions of Bernoulli's equation. Next two results indicate that a solution of Bernoulli's equation can be considered as a solution of the Briot-Bouquet equation. A more interesting outcome is that the equation has a positive real and univalent solution.

Corollary 3.5 Consider Bernoulli's equation
Then the solution is defined by the hypergeometric function as follows: where c is a nonzero constant.

Conclusion
The above study showed a deep investigation of a class of analytic normalized functions in the open unit disk taking Bernoulli's formula equations. The class was studied previously when β < α ≤ 1 (see [14]). In this effort, we considered α ≤ 1 ≤ β < ∞. Applications are illustrated to discover the geometric behavior of solutions, not only for the complex Bernoulli's equations, but for some inequalities as well. The recent work can be studied by suggesting other classes of analytic functions such as meromorphic [17], harmonic, and multivalent classes [18].