PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 101(115) (2017), 143–149 https://doi.org/10.2298/PIM1715143K

A NOTE ON THE FEKETE–SZEGÖ PROBLEM

FOR CLOSE-TO-CONVEX FUNCTIONS WITH RESPECT TO CONVEX FUNCTIONS

Bogumiła Kowalczyk, Adam Lecko, and H. M. Srivastava

Abstract. We discuss the sharpness of the bound of the Fekete–Szegö functional for close-to-convex functions with respect to convex functions. We also briefly consider other related developments involving the Fekete–Szegö functional |a_{3} − λa_{2}^{2}| (0 ⩽ λ ⩽ 1) as well as the corresponding Hankel determinant for the Taylor–Maclaurin coefficients {a_{n} }_{n∈ℕ∖{1}} of normalized univalent functions in the open unit disk 𝔻, ℕ being the set of positive integers.

1. Introduction

A classical problem in geometric function theory of complex analysis, which was settled by Fekete and Szegö [4], is to find for each λ ∈ [0, 1] the maximum value of the coefficient functional Φ_{λ}(f) given by

(1.1)  Φ_{λ}(f) := |a_{3} − λa_{2}^{2} |

over the class S of univalent functions f in the open unit disk

𝔻 := {z : z ∈ ℂ and |z| < 1}

of the following normalized form (see, for details, [5, 22, 24]):

(1.2)  f(z) = z + ∑_{n=2}^{∞} a_{n}z^{n}  (z ∈ 𝔻).

By applying the Loewner method, Fekete and Szegö [4] proved that

max_{f∈S}Φ_{λ}(f) = \{ \begin{matrix}1 + 2 exp(− \frac{2λ}{1−λ} ) & (0 ⩽ λ < 1) \\ 1 & (λ = 1). \end{matrix}

2010 Mathematics Subject Classification: Primary 30C45. Key words and phrases: analytic functions, convex functions, Fekete–Szegö problem, Hankel

determinant, Taylor–Maclaurin coefficients, close-to-convex functions with respect to a convex function, Carathéodory class, Schwarz functions.

Communicated by Stevan Pilipović.

For various compact subclasses F of the class A of all analytic functions f in 𝔻 of the form (1.2), as well as with λ being an arbitrary real or complex number, many authors computed

(1.3)  max_{f∈F}Φ_{λ}(f)

or calculated the upper bound of (1.3) (see, e.g., [2, 8, 11, 21]).

Let S^{∗} denote the class of starlike functions, that is, f ∈ S^{∗} if

f ∈ A and Re( \frac{zf′(z)}{f(z)} ) > 0 (z ∈ 𝔻).

Given δ ∈ (−\frac{π}{2} , \frac{π}{2} ) and g ∈ S^{∗}, let C_{δ}(g) denote the class of functions called closeto-convex with argument δ with respect to g, that is, the class of all functions f ∈ A such that

(1.4)  Re( e^{iδ} \frac{zf′(z)}{g(z)} ) > 0 (z ∈ 𝔻).

We also suppose that, given g ∈ S^{∗}, C(g) := ⋃ _{g∈S^{∗}} C_{δ}(g) and that, given δ ∈ (−\frac{π}{2} , \frac{π}{2} ), C_{δ} := ⋃ _{g∈S^{∗}} C_{δ}(g). Let

C := ⋃_{δ∈(− \frac{π}{2} , \frac{π}{2} )} ⋃_{g∈S^{∗}} C_{δ}(g)

denote the class of close-to-convex functions (see, for details, [20, pp. 184–185], [6, 10]).

For the whole class C, the sharp bound of the Fekete–Szegö coefficient functional Φ_{λ} for λ ∈ [0, 1], given by (1.1), was calculated by Koepf [13] who extended the earlier result for the class C_{0} and for λ ∈ ℝ due to Keogh and Merkes [11], namely, it holds

max_{f∈C}Φ_{λ}(f) = max_{f∈C_{0}}Φ_{λ}(f) = \{ \begin{matrix}|3 − 4λ| & (λ ∈ ( − ∞, \frac{1}{3}] ∪ [1, ∞)) \\ \frac{1}{3} + \frac{4}{9λ} & (λ ∈ [ \frac{1}{3}, \frac{2}{3}]) \\ 1 & (λ ∈ [ \frac{2}{3}, 1]). \end{matrix}

For various subclasses of the class of close-to-convex functions, the problem to estimate the coefficient functional Φ_{λ} is continued in several subsequent works (see, for details, [9, 12, 14–16]). Some interesting and important subclasses of the class C are the classes C_{δ}^{c} and C^{c}, which are defined below.

Let S^{c} denote the class of convex functions, that is, f ∈ S^{c} if

f ∈ A and Re( 1 + \frac{zf′′(z)}{f′(z)} ) > 0 (z ∈ 𝔻).

Since S^{c} ⊊ S^{∗}, the class C_{δ}^{c} := ⋃ _{g∈S^{c}} C_{δ}(g) is a proper subclass of the class C_{δ} and the class

C^{c} := ⋃_{δ∈(− \frac{π}{2} , \frac{π}{2} )} ⋃_{g∈S^{c}} C_{δ}(g)

is a proper subclass of the class C.

The class C_{0}^{c} was defined by Abdel-Gawad and Thomas [1]. The class C^{c} of close-to-convex functions with respect to convex functions was introduced by Srivastava, Mishra and Das [23]. In both of these cited papers, the authors (Abdel-Gawad and Thomas [1] and Srivastava, Mishra and Das [23]) considered the coefficient functional Φ_{λ} with λ ∈ [0, 1] also. In fact, in Srivastava, Mishra and Das [23] extended, for the class C^{c}, the earlier result of Abdel-Gawad and Thomas [1] for the class C_{0}^{c}. However, in each of the above-cited papers, the proof for the sharpness of the bound in (1.3) for λ ∈ ( \frac{2}{3}, 1] was proposed incorrectly as 5/6.

This note is motivated essentially by the earlier papers [1] and [23]. The main purpose of our investigation here is to discuss such sharpness results for the bound in (1.3). We also provide a rather brief consideration of other related developments involving the Fekete–Szegö functional |a_{3} − λa_{2}^{2} | (0 ⩽ λ ⩽ 1) in (1.1) as well as the corresponding Hankel determinant for the Taylor–Maclaurin coefficients {a_{n}}_{n∈ℕ∖{1}} of normalized univalent functions of the form (1.2).

2. Main Observation

As we remarked in Section 1, in both of the afore cited papers [1,23], the upper bounds of the Fekete–Szegö coefficient functional Φ_{λ} (0 ⩽ λ ⩽ 1) for the classes C_{0}^{c} and C^{c}, were computed. In fact, Theorems 5 and 6 of Srivastava, Mishra and Das [23] state that the following sharp inequality

(2.1)  max_{f∈C^{c}}Φ_{λ}(f) ⩽ \frac{5}{6} (λ ∈ [ \frac{2}{3}, 1])

holds true and that this result is the same as in [1] for the class C_{0}^{c} (a part of Theorem 3). However, the assertion that the extremal function, for which the equality in (2.1) is satisfied when λ ∈ (\frac{2}{3}, 1], belongs to C^{c} is incorrect. Indeed, here in this section, we note that the above-cited papers [1, 23] contain a statement to the effect that the equality in (2.1) is attained by a function f ∈ A given by

(2.2)  \frac{zf′(z)}{h(z)} = \frac{1 + ω(z)}{1 − ω(z)} (z ∈ 𝔻),

where h ∈ S^{c} is of the form

(2.3)  h(z) = z + ∑_{n=2}^{∞} b_{n}z^{n} (z ∈ 𝔻; b_{2} = b_{3} := 1)

and ω is a function of the form

(2.4)  ω(z) = ∑_{n=1}^{∞} β_{n}z^{n} (z ∈ 𝔻)

with

(2.5)  β_{1} := \frac{2 − 3λ}{6λ} ± i\frac{\sqrt{6λ − 4}}{6λ}  and β_{2} := 1 − β_{1}^{2}.

Unfortunately, however, ω is not a Schwarz function for λ ∈ (\frac{2}{3}, 1]. We recall here that a Schwarz function means an analytic self-mapping of 𝔻 with ω(0) := 0. Let us denote the class of Schwarz functions by B_{0}. In order to see that ω ∉ B_{0}, we verify (by straightforward computation) that, for λ ∈ (\frac{2}{3}, 1], the following inequality:

(2.6)  |β_{2}| ⩽ 1 − |β_{1}|^{2}

is false, so a necessary condition for ω to be in B_{0} (see, for example, [5, Vol. II, p. 78]) does not hold true. Alternatively, in order to get a contradiction, we suppose that ω with its coefficients in (2.5) is a Schwarz function. Thus, clearly, (2.6) holds true. Hence we find from (2.5) that 1 − |β_{1}|^{2} ⩾ |β_{2}| = |1 − β_{1}^{2}| ⩾ 1 − |β_{1}|^{2}. Thus we have |1 − β_{1}^{2}| = 1 − |β_{1}|^{2} and, therefore, β_{1} = |β_{1}| or β_{1} = −|β_{1}|. This means that β_{1} is a real number, which by (2.5) is possible only for λ = \frac{2}{3}. Consequently, for λ ∈ (\frac{2}{3}, 1], the function ω with its coefficients in (2.5) does not belong to B_{0}. So, in light of (2.2), it does not follow that f is in C^{c} or in C_{0}^{c}.

Equivalently, let

(2.7)  p(z) := \frac{1 + ω(z)}{1 − ω(z)} (z ∈ 𝔻),

where ω is as given above. Then

(2.8)  p(z) = 1 + ∑_{n=1}^{∞} c_{n}z^{n} (z ∈ 𝔻),

where, in view of (2.7), (2.4) and (2.5), we have c_{1} = 2β_{1} and c_{2} = 2(β_{2} + β_{1}^{2}) = 2. We observe further that, for λ ∈ (\frac{2}{3}, 1], the function p does not belong to the Carathéodory class. We recall here that the Carathéodory class, denoted as P, contains analytic functions p of the form (2.8) with a positive real part. In order to see that p ∉ ℙ, we verify for λ ∈ (\frac{2}{3}, 1] that the inequality |c_{2} − c_{1}^{2}/2| ⩽ 2 − |c_{1}|^{2}/2, is false, which happens to be a necessary condition for p to be in the class ℙ (see, for example, [22, p. 166]).

3. Concluding remarks and further developments

By means of Theorem 3 of Abdel-Gawad and Thomas [1], Theorems 1 to 4 of Srivastava, Mishra and Das [23], and in light of our observation in Section 2, we arrive at the following result.

Theorem 1. Each of the following assertions holds true:

(3.1)  max_{f∈C^{c}}Φ_{λ}(f) = max_{f∈C_{0}^{c}}Φ_{λ}(f) = \{ \begin{matrix}\frac{5}{3} − \frac{9λ}{4} & (λ ∈ [0, \frac{2}{9}]) \\ \frac{2}{3} + \frac{1}{9λ} & (λ ∈ [ \frac{2}{9}, \frac{2}{3}]) \end{matrix}
(3.2)  max_{f∈C^{c}}Φ_{λ}(f) ⩽ \frac{5}{6}  (λ ∈ ( \frac{2}{3}, 1]).

Remark 1. The sharpness of the inequality in (3.2) for the classes C^{c} and C_{0}^{c} is an open problem.

We now note that, by Loewner Theorem (see, for example, [5, Vol. I, p. 1127]), the function h ∈ S^{c} of the form (2.3) (with b_{2} = b_{3} := 1) is uniquely determined, that is, h(z) = \frac{z}{1−z} = ∑ _{n=1}^{∞} z^{n} (z ∈ 𝔻). Then (1.4) with g := h is of the form

(3.3)  Re(e^{iδ}(1 − z)f′(z)) > 0  (z ∈ 𝔻)

and defines the class C_{δ}(h), and further the class C(h). For the first time, the inequality in (3.3), treated as the univalence criterion, was distinguished explicitly in [20, p. 185]. For the class C(h), the upper bound of the Fekete–Szegö coefficient functional Φ_{λ} for λ ∈ ℝ was recently obtained in [14], where the following result was proven.

Theorem 2. It is asserted that

(3.4) max_{f∈C(h)} Φ_{λ}(f) ⩽ \{ \begin{matrix}| \frac{1}{3} − \frac{1}{4}λ| + \frac{2}{3} |2 − 3λ| & (λ ∈ ( − ∞, \frac{2}{9}] ∪ [ \frac{10}{9} , ∞)) \\ \frac{1}{12} · \frac{(2−3λ)^{2}}{2−|2−3λ|} + | \frac{1}{3} − \frac{1}{4}λ| + \frac{2}{3} & (λ ∈ [ \frac{2}{9}, \frac{10}{9} ]). \end{matrix}

For each λ ∈ ( − ∞, \frac{2}{3}] ∪ [ \frac{4}{3}, ∞), the inequality is sharp and the equality in (2) is attained by a function in C_{0}(h).

Remark 2. For λ ∈ ( − ∞, \frac{2}{3}] ∪ [ \frac{4}{3}, ∞), we can rewrite (3.4) as the following

corollary.

Corollary 1. The following assertion holds true:

(3.5)  max_{f∈C(h)} Φ_{λ}(f) = \{ \begin{matrix}| \frac{5}{3} − \frac{9λ}{4} | & (λ ∈ ( − ∞, \frac{2}{9}] ∪ [ \frac{4}{3}, ∞)) \\ \frac{2}{3} + \frac{1}{9λ} & (λ ∈ [ \frac{2}{9}, \frac{2}{3}]). \end{matrix}

Remark 3. For λ ∈ [0, \frac{2}{3}], the result (3.5) asserted by Corollary 3.5 coincides with (3.1). Thus, naturally, Theorem 1 and Theorem 2 yield Corollary 2 below.

Corollary 2. Each of the following assertions holds true:

max_{f∈C(h)} Φ_{λ}(f) = max_{f∈C_{0}^{c}}Φ_{λ}(f) = max_{f∈C^{c}}Φ_{λ}(f) (λ ∈ [0, \frac{2}{3}]),

max_{f∈C(h)} Φ_{λ}(f) ⩽ \frac{9λ^{2} − 30λ + 26}{6(4 − 3λ)} ⩽ \frac{5}{6} (λ ∈ ( \frac{2}{3}, 1]).

Remark 4. The maximum of Φ_{λ} for λ ∈ [0, \frac{2}{3}], over the class C^{c} of closeto-convex functions with respect to convex functions and over its subclass C(h) of close-to-convex functions with respect to convex function h, are identical.

Remark 5. The sharpness of the inequality in (3.4) for λ ∈ ( \frac{2}{3}, \frac{4}{3}) is an open problem.

Remark 6. We reiterate the fact that the Fekete–Szegö coefficient functional |a_{3} − λa_{2}^{2} | is well known for its rich history in geometric function theory. Its origin was in the disproof by Fekete and Szegö [4] of the 1933 conjecture of Littlewood and Paley that the coefficients of odd univalent functions are bounded by unity (see, for details, [4]). The λ-generalized Fekete–Szegö coefficient functional |a_{3} − λa_{2}^{2} | has since received great attention, particularly in connection with many subclasses of the class S of normalized analytic and univalent functions. On the other hand, in the year 1976, Noonan and Thomas [17] defined the 𝔮th Hankel determinant of the function f in (1.2) by

H_{𝔮}(n) = |\begin{matrix}a_{n} & a_{n+1} & ⋯ & a_{n+𝔮−1} \\ a_{n+1} & a_{n+2} & ⋯ & a_{n+𝔮} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ a_{n+𝔮−1} & a_{n+𝔮} & ⋯ & a_{n+2𝔮−2} \end{matrix} | (n, 𝔮 ∈ ℕ; a_{1} := 1).

The determinant H_{𝔮}(n) has also been considered by several other authors. For example, Noor [18] determined the rate of growth of H_{𝔮}(n) as n → ∞ for functions f given by (1.1) with bounded boundary. In particular, sharp upper bounds on H_{2}(2) were obtained in the recent works [7, 18] for different classes of functions. We note, in particular, that

H_{2}(1) = |\begin{matrix}a_{1} & a_{2} \\ a_{2} & a_{3} \end{matrix} | = a_{3} − a_{2}^{2} and H_{2}(2) = |\begin{matrix}a_{2} & a_{3} \\ a_{3} & a_{4} \end{matrix} | = a_{2}a_{4} − a_{3}^{2}.

The Hankel determinant H_{2}(1) = a_{3} − a_{2}^{2} is the classical Fekete–Szegö coefficient functional. The upper bounds of H_{2}(2) for some specific analytic function classes were discussed quite recently by Deniz et al. [3] (see also [19]).

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Faculty of Mathematics and Computer Science (Received 18 06 2016) Department of Complex Analysis (Revised 12 11 2016) University of Warmia and Mazury Olsztyn Poland

b.kowalczyk@matman.uwm.edu.pl
alecko@matman.uwm.edu.pl

Department of Mathematics and Statistics University of Victoria Victoria Canada; China Medical University Taichung Taiwan Republic of China

harimsri@math.uvic.ca