Third-Order Hankel Determinant for Certain Class of Analytic Functions Related with Exponential Function

Received: 30 July 2018; Accepted: 9 October 2018; Published: 15 October 2018 Abstract: Let S∗ l denote the class of analytic functions f in the open unit disk D = {z : |z| < 1} normalized by f (0) = f ′(0)− 1 = 0, which is subordinate to exponential function, z f ′(z) f (z) ≺ e z (z ∈ D). In this paper, we aim to investigate the third-order Hankel determinant H3(1) for this function class S∗ l associated with exponential function and obtain the upper bound of the determinant H3(1). Meanwhile, we give two examples to illustrate the results obtained.

( Assume that P denote the class of analytic functions p normalized by and satisfying the condition Re p(z) > 0 (z ∈ D).
It is easy to see that, if p(z) ∈ P, then exists a Schwarz function ω(z) with ω(0) = 0 and |ω(z)| < 1, such that (see [1]) Now, we start with recalling the definition of subordination.Suppose that f and g are two analytic functions in D.Then, we say that the function g is subordinate to the function f , and we write g(z) ≺ f (z) (z ∈ D), if there exists a Schwarz function ω(z) with ω(0) = 0 and |ω(z)| < 1, such that (see [2]) Recently, Mendiratta et al. in [3] introduced the following subclass S * l of analytic functions associated with exponential function.Definition 1. (see [3]).A function f ∈ S is said to be in the class S * l , if it satisfies the following condition: We easily observe that, f ∈ S * l , if and only if In fact, if we choose f (z) = z + 1 4 z 2 , then, from Equation (3), we can sketch the figure of the function class S * l (see Figure 1).The q th Hankel determinant for q ≥ 1 and n ≥ 1 is stated by Noonan and Thomas [4] as This determinant has been considered by several authors, for example, Noor [5] determined the rate of growth of H q (n) as n → ∞ for functions f (z) given by Equation (1) with bounded boundary and Ehrenborg [6] studied the Hankel determinant of exponential polynomials.
In recent years, many authors studied the second-order Hankel determinant H 2 (2) and the third-order Hankel determinant H 3 (1) for various classes of functions, the interested readers can see, for example, [13][14][15][16][17][18][19][20][21][22].We note that, they discussed the determinants H 2 (2) and H 3 (1) based on the function classes, which are all subordinate to a certain function 1+Az 1+Bz (−1 ≤ B < A ≤ 1; z ∈ D).Until now, very few researchers have studied the above determinants for the function class, subordinated to e z (z ∈ D).So, in this paper, we aim to investigate the third-order Hankel determinant H 3 (1) for the function class S * l , which is associated with exponential function, and obtain the upper bound of the above determinant.

Main Results
In order to prove our desired results, we shall require the following lemmas.Lemma 1. (see [23]).If p(z) ∈ P, then exists some x, z with |x| ≤ 1, |z| ≤ 1, such that Lemma 2. (see [24]).Let p(z) ∈ P, then (see [3]).If the function f (z) ∈ S * l and of the form Equation (1), then We now state and prove the main results of our present investigation.
Theorem 1.If the function f (z) ∈ S * l and of the form Equation ( 1), then we have Proof.Since f (z) ∈ S * l , according to the definition of subordination, then there exists a Schwarz function ω(z) with ω(0) = 0 and |ω(z Define a function Then, we notice that p(z) ∈ P and On the other hand, On comparing the coefficients of z, z 2 , z 3 between the Equations ( 6) and ( 7), we obtain So, Using Lemma 1, we thus know that and applying the triangle inequality, the above equation reduces to Suppose that which shows that F(c, t) is an increasing function on the closed interval [0,1] about t.Therefore, the function F(c, t) can get the maximum value at t = 1, that is Next, let Then, we easily find the function G(c) have a maximum value at c = 0, also which is The proof of Theorem 1 is thus completed.
Theorem 2. If the function f (z) ∈ S * l and of the form Equation ( 1), then we have Proof.From the Equation ( 8), we have Again, by applying Lemma 1, we get . Then, using the triangle inequality, we deduce that Let G (c) = 0, then the root is c = r = −4+8 √ 2 7 . And so the function G(c) have a maximum value , also which is The proof of Theorem 2 is completed.
Theorem 3. If the function f (z) ∈ S * l and of the form Equation ( 1), then we have Proof.Suppose that f (z) ∈ S * l , then from Equation ( 8), we have In view of Lemma 1, we thus obtain S denote the class of functions f which are analytic and univalent in the open unit disk D = {z : |z| < 1} of the form f (z) = z + ∞ ∑ n=2 a n z n (z ∈ D).

Figure 1 .
Figure 1. the figure of the function class S * l for f (z) = z + 1 4 z 2 .