Estimates of Coefficient Functionals for Functions Convex in the Imaginary-Axis Direction

Let C0(h) be a subclass of analytic and close-to-convex functions defined in the open unit disk by the formula Re{(1−z2)f′(z)}>0. In this paper, some coefficient problems for C0(h) are considered. Some properties and bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates of the difference and of sum of successive coefficients, bounds of the sum of the first n coefficients and bounds of the n-th coefficient. The obtained results are used to determine coefficient estimates for both functions convex in the imaginary-axis direction with real coefficients and typically real functions. Moreover, the sum of the first initial coefficients for functions with a positive real part and with a fixed second coefficient is estimated.


Introduction
Coefficient problems of analytic functions have always been of the great interest to researchers. Let A be a class 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}. There are many papers in which the n-th coefficient a n has been estimated in various subclasses of analytic functions. The difference of the moduli of successive coefficients |a n+1 | − |a n | of a certain class of functions was also estimated (see, for example, [1][2][3][4]). The idea of estimating the difference of successive coefficients |a n+1 − a n | follows from the obvious inequality Let us start with the notation and the definitions. By P we denote the class of analytic functions q with a positive real part in ∆, having the Taylor series expansion q(z) = 1 + ∞ ∑ n=1 p n z n . (2) A subclass of P consisting of functions with real coefficients is denoted by P R . Let T denote the class of typically real functions, i.e., functions f ∈ A which satisfy the condition Im{z} · Im{ f (z)} ≥ 0 for all z ∈ ∆. All coefficients of any f ∈ T are real. This results in the symmetry of f (∆) with respect to the real axis. It is worth recalling that there exists a unique correspondence between the functions in T and P R (see, [8]) Let S * denote the class of starlike functions, i.e., functions f ∈ A such that Re{z f (z)/ f (z)} > 0 for all z ∈ ∆. Given that β ∈ (−π/2, π/2) and g ∈ S * , a function f ∈ A is called close-to-convex with argument β with respect to g if Re The class of all functions satisfying (3) is denoted by C β (g) (see [9]). Coefficient problems for the class C 0 (k), where k is the Koebe function k(z) = z/(1 − z) 2 , were discussed in a few papers (see, for example, [10][11][12][13]). In this paper we consider the or equivalently, where q is in P. Directly from the properties of P, it follows that C 0 (h) is a convex family, i.e., α f + (1 − α)g ∈ C 0 (h) providing that f , g ∈ C 0 (h) and α ∈ [0, 1]. Moreover, the following property of symmetry is valid in . It is clear thatf (∆) and f (∆) are mutually symmetric with respect to the real axis. Other important properties of C 0 (h) are given in the three following theorems (see, [14]).
Theorem 2. If f ∈ C 0 (h), then f is convex in the direction of the imaginary axis.

Theorem 3.
Let all coefficients of f given by (1) be real. Then, In the above, K R (i) denotes the class of functions of the Form (1) which are convex in the direction of the imaginary axis and have all real coefficients. Robertson [15] proved that From (5), it follows that na n = p 1 + p 3 + . . . + p n−1 if n is even (7) and where p n are the coefficients of functions from the class P.
In this paper we find bounds of different functionals depending on the second coefficient a 2 of f ∈ C 0 (h). In fact, it is more convenient to express our results in terms of p = p 1 , applying the correspondence To make the results more legible, we define the class C 0 (h, p), p ∈ [−2, 2] as follows Observe that in two particular cases: when p = −2 or p = 2, the class C 0 (h, p) consists of only one function. Namely, if S n = 1 + a 2 + a 3 + . . . + a n .
Moreover, for q ∈ P given by (2) we define

Auxiliary Lemmas
In order to prove our results, we need a few lemmas concerning functions in the class P. The first one is known as Caratheodory's lemma (see, for example, [16]). The second one is due to Hayami and Owa ([17]) and the third one is the result of Libera and Złotkiewicz ( [18,19]).
The next two lemmas relate to the bounds of functionals P n and Q n defined in (12) and (13).

Lemma 6.
If q ∈ P is given by (2) and p = p 1 ∈ R, then and Equality holds for Function (14), if n is even and p ∈ [0, 2], and if p = −2 and p = 2 for all positive integers n.
Proof. Let n be even. We can write Applying Formula (17) we obtain This proves (18). Let n be odd. Then which is equivalent to (19).
It is easy to check that P n = n + 1 2 np, if h is given by (14), n is even and p ∈ [0, 2]. Equalities for the cases p = −2 and p = 2 are also easy to verify.
If we take q(−z) instead of q(z) in Lemma 6, then we obtain the estimate of Q n .

Lemma 7.
If q ∈ P is given by (2) and p = p 1 ∈ R, then and Equality holds for Function (14), if n is even and p ∈ [−2, 0] and if p = −2 and p = 2 for all positive integers n.
The last lemma is a special case of a more general result due to Choi, Kim and Sugawa [22].

Bounds of |F n |
In this section we estimate the difference of successive coefficients for f ∈ C 0 (h, p). The functional F n is defined in (9).
In the following theorem we derive the sharp bounds of |F 2 | and |F 3 | for f ∈ C 0 (h, p) and each p ∈ [−2, 2].

Bounds of |G n |
In this section, we find estimates of the functional G n defined in (10). If f ∈ C 0 (h, p), then from (7) and (8) it follows that G n = (−1) n (1 − Q n ) .
This result can be improved for even n, if we rearrange G n as follows Hence, from Formulas (16) and (17), Combining (28) and Lemma 7, we can formulate the main theorem of this section.
Apart of the sharp bounds of |G n | for even n, we can also derive the sharp bound of G 3 .

Bounds of |S n |
In this section we determine the bounds of the functional S n defined in (11), i.e., we find estimates of the sum of the first n coefficients of f ∈ C 0 (h, p). To prove the main theorem of this section, we use the following three theorems. The proof of the first one is analogous to the proof of Theorem 4.
Proof. If n is odd, n ≥ 5, then from (11) we have Taking into account Theorems 8 and 9, for n ≥ 5 we get If n is even, n ≥ 6, then from (11) we have Taking into account Theorem 8, we obtain By applying Theorem 10 and making a simple calculation, we obtain the desired estimate of S n for even n.
Taking p = 2 in Theorem 11, we obtain the sharp bound |S n | ≤ n. The sharpness of this result is a simple consequence of the sharpness of Theorems 8-10.

Bounds of |a n |
In all results presented above, the estimates of the functionals defined for functions f ∈ C 0 (h) depend on the fixed second coefficient. Consequently, the natural question arises about the bound of the n-th coefficient.
Observe that the function f given by (1) has all even coefficients equal to 1, independently the second coefficient. Therefore, we may pose a question about the bounds of odd coefficients when a 2 is fixed or the bounds of all coefficients under the assumption that a 3 is fixed. We shall give the answer to the second question provided that a 3 is a real number.
We need the lemma which is a simple consequence of the set of variability of (p 1 , p 2 ), where p 1 and p 2 are the coefficients of a function q ∈ P R . Lemma 9. If q ∈ P R , then p 1 2 − 2 ≤ p 2 ≤ 2.
In view of this lemma, we immediately get that, if f ∈ C 0 (h) and a = a 3 ∈ R, then a ∈ [−1/3, 1]. Now, we are ready to derive the bound of a n for f ∈ C 0 (h).

Applications for Typically Real Functions
From Theorem 3 it follows that K R (i) ⊂ C 0 (h). Therefore, all estimates obtained in Sections 3-6 are also valid for K R (i). These bounds are sharp for any fixed p in [−2, 2] and for n = 2 (Formulae (23), (29) and (33)), for n = 3 (Formulae (24), (31) and (34)) and for n = 4 (Formula (35)). One of the results, namely (29), is sharp even for all positive even integers n. In the majority of cases, the extremal functions are those given by (25) and by (26).
The application of the relation between K R (i) and T and the results found in Section 4 lead to obtaining the estimates of successive coefficients for typically real functions. These results would be difficult to obtain in any other way.
Let g(z) = z f (z). From (6), we know that f ∈ K R (i), if and only if, g ∈ T . If f is of the Form (1) and g(z) = z + b 2 z 2 + . . . , then na n = b n .
Consequently, we obtain the following two corollaries.