Hankel Determinants for Univalent Functions Related to the Exponential Function

Recently, two classes of univalent functions S e * and K e were introduced and studied. A function f is in S e * if it is analytic in the unit disk, f ( 0 ) = f ′ ( 0 ) − 1 = 0 and z f ′ ( z ) f ( z ) ≺ e z . On the other hand, g ∈ K e if and only if z g ′ ∈ S e * . Both classes are symmetric, or invariant, under rotations. In this paper, we solve a few problems connected with the coefficients of functions in these classes. We find, among other things, the estimates of Hankel determinants: H 2 , 1 , H 2 , 2 , H 3 , 1 . All these estimates improve the known results. Moreover, almost all new bounds are sharp. The main idea used in the paper is based on expressing the discussed functionals depending on the fixed second coefficient of a function in a given class.


Introduction
Let A be the collection of functions of the form f (z) = z + ∞ ∑ n=2 a n z n (1) which are analytic in the open unit disk ∆ = {z ∈ C : |z| < 1} and let S denote the subclass of A consisting of functions which are univalent in ∆.
Since the early twentieth century many mathematicians have been interested in different problems involving the coefficients of functions f in a given subclass of A. The most important and inspiring problem known as the Bieberbach conjecture was solved by de Branges only 70 years after its formulation. Over the years, many interesting tasks connected with these coefficients appeared. The most important ones were settled by Robertson, Bombieri, Zalcman, Krzyż and Landau. In the 1960s Pommerenke defined the Hankel determinant H k,n , for a given f of the form (1), as follows H k,n = a n a n+1 . . . a n+k−1 a n+1 a n+2 . . . a n+k · · · · · · · · · · · · a n+k−1 a n+k . . . a n+2k−2 , ( where n, k ∈ N. The studies on Hankel determinants are concentrated on estimating H 2,2 and H 3,1 for different subclasses of S. These particular determinants can be written as and Let us recall that for two functions f and g analytic in ∆, we say that g is subordinated to f (g ≺ f ), if there exists a function ω analytic in ∆ with ω(0) = 0, |ω(z)| < 1 and such that g(z) = f (ω(z)).
Let φ be an analytic function such that φ(z) > 0 for z ∈ ∆ with φ(0) = 1. Ma and Minda [4] defined the classes of S * (φ) and K(φ) in the following way and From (5) it is seen that S * (φ) reduces to S * if φ(z) = 1+z 1−z . For other specific choices of φ we obtain for example: , then S * (φ 2 ) = S * α is the class of strongly starlike functions of order α, 3.
In a similar way we defined the relative subclasses of K.
Recently, Mendiratta et al. [5] discussed S * e and K e , i.e., the classes S * (φ) and K(φ) with φ(z) = e z . Various problems, including distortion and growth theorems, radii problems, inclusion relations and coefficient estimates, were discussed there.
In their two following papers Zhang et al. [6] and Shi et al. [7] broadened the range of discussed coefficient problems. They found the coefficient estimates for a n , n = 2, 3, 4, 5 and the bounds of the following functionals: a 2 a 3 − a 4 , H 2,1 , H 2,2 and, as a consequence, H 3,1 . Except for the bounds of a n , n = 2, 3, 4 and H 2,1 , all results are non-sharp.
In this paper, we improve all non-sharp results mentioned above. The main idea is to express the discussed functionals depending on the second coefficient of f ∈ S * e or g ∈ K e . In fact, the second coefficient of f ∈ S * e or g ∈ K e can be replaced by the coefficient p 1 of a corresponding function P with a positive real part. This idea leads to better estimates than those in [6,7]. Moreover, the new bounds of a 2 a 3 − a 4 and H 2,2 are sharp.

Auxiliary Lemmas
To prove our results, we need two lemmas concerning functions in the class P of functions P such that P(z) > 0 and P has the Taylor series representation Lemma 1 ([8]). Let p 1 ∈ [0, 2]. A function P of the form (9) is in P if and only if 1.
for some x and y such that |x| ≤ 1, |y| ≤ 1. 4,9]). If P ∈ P is of the form (9) and µ ∈ [0, 1], then the following sharp estimates hold In fact, the second inequality of Lemma 2 did not appear in [4], but it is a reformulation of the result obtained for bounded functions.
At the end of this section, observe that both classes S * e and K e possess a specific type of symmetry.

They are invariant (or symmetric) under rotations. Recall that the class A is invariant under rotations when f is in
It can be easily checked that S * e and K e , as well as the functionals |a 2 a 3 − a 4 |, |H 2,1 |, |H 2,2 | and |H 3,1 | considered in S * e or in K e , satisfy the above definitions. Due to the symmetry described above, in the considerations we can assume that one coefficient (usually the second one) is a positive real number.

Coefficient Problems for S * e
It follows from the definition of S * e that there exists a function ω(z) with Define P(z) = 1+ω(z) 1−ω(z) = 1 + p 1 z + p 2 z 2 + . . .. The function P is in P and (10) is equivalent to Now, expanding both sides of (11) in the Taylor series and comparing coefficients at z k , k = 1, 2, 3, 4, we obtain (see also Formulae (15)-(18) in [7]), Now, we can prove the following theorem.
Proof. The bound of a 3 is clear. To obtain the bound of a 4 and a 5 it is enough to write and to apply Lemma 2.
Corollary 1. If f ∈ S * e is given by (1), then 1. The function which gives equality in the bounds of a 2 , a 3 and a 4 corresponds to P(z) = 1+z 1−z , so ω(z) = z. This means that the extremal function is of the form so In [5], Mendiratta et al. proved that if f ∈ S * e , then |a 3 − a 2 2 | ≤ 1 2 . Although this inequality is sharp, we can easily generalize it by applying Lemma 2 in the following identity a 3 − a 2 2 = 1 4 (p 2 − 3 4 p 1 2 ).
For sharpness, it is enough to discuss a function f which corresponds to For this function, Consequently, we get the following corollary.
Corollary 2. If f ∈ S * e is given by (1), then The result is sharp.
Consequently, we get the following corollary.
Corollary 3. If f ∈ S * e is given by (1), then The bound is sharp.
Observe that the equality in (20) holds only when x = −1. This means that P is of the form (15). Hence, For p = 0 we have P 0 (z) = 1−z 2 1+z 2 , so from (11), Hence, the corresponding function in S * e is of the form Finally, we find a new bound of H 3,1 for the class S * e . In [6] it was proved that |H 3,1 | ≤ 0.565. . . In the succeeding paper Shi et al. showed that |H 3,1 | ≤ 0.500 . . . (see, Theorem 1 in [7]). We improve these results essentially in the following way.

Coefficient Problems for K e
Directly from the definitions of S * e and K e it follows that for f (z) = zg (z), Consequently, if g ∈ K e , g(z) = z + b 2 z 2 + . . .
and f ∈ S * e is given by (1), then b n = a n /n. For this reason, Theorem 1 results in the two following facts.
The proof of this theorem is obvious. Theorem 8. If g ∈ K e is given by (27) and |b 2 | = p/4, p ∈ [0, 2], then Consequently, we get the following corollary.
Corollary 5. If g ∈ K e is given by (27), then The result is sharp.
Hence f 2 (z) = z · exp The bound is sharp.
Consequently, we get the following corollary.
Although this consequently gives |a 5 | ≤ 13 44 for f ∈ S * e and |b 5 | ≤ 13 220 for g ∈ K e , the final bounds of H 3,1 for S * e and K e are only slightly better than these from Theorem 5 and Theorem 10.