Upper Bound of Second Hankel Determinant for Certain Subclasses of Analytic Functions

Abstract

In this present investigation, we first give a survey of the work done so far in this area of Hankel determinant for univalent functions. Then the upper bounds of the second Hankel determinant |𝑎₂𝑎₄−𝑎²₃| for functions belonging to the subclasses 𝑆(𝛼,𝛽), 𝐾(𝛼,𝛽), 𝑆^∗_𝑠(𝛼,𝛽), and 𝐾_𝑠(𝛼,𝛽) of analytic functions are studied. Some of the results, presented in this paper, would extend the corresponding results of earlier authors.

1. Introduction

Let 𝒜 denote the class of functions of the form 𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘 (1) which are analytic in the unit disc 𝕌={𝑧∶|𝑧|<1}, and let 𝑆 denote the subclass of 𝒜 that is univalent in 𝕌. Suppose that 𝑓 and 𝑔 are analytic functions in 𝕌; we say that 𝑓 is subordinate to 𝑔, written 𝑓≺𝑔, if there exists a Schwarz function 𝜔, which is analytic in 𝕌 with 𝜔(0)=0 and |𝜔(𝑧)|<1 for all 𝑧∈𝕌, such that 𝑓(𝑧)=𝑔(𝜔(𝑧)), 𝑧∈𝕌. In particular, if 𝑔 is univalent in 𝕌, then the subordination is equivalent to 𝑓(0)=𝑔(0) and 𝑓(𝕌)⊂𝑔(𝕌).

Let 𝒫 be the family of all functions 𝑝 analytic in 𝕌 for which ℜ{𝑝(𝑧)}>0 and 𝑝(𝑧)=1+𝑐₁𝑧+𝑐₂𝑧²+⋯ (2) for 𝑧∈𝕌.

It is well known that the following correspondence between the class 𝒫 and the class of Schwarz functions 𝜔 exists [1]: 𝑝∈𝒫⟺𝑝=1+𝜔.1−𝜔 (3)

Let 𝑆^∗ denote the starlike subclass of 𝑆. It is well known that 𝑓∈𝑆^∗ if and only if ℜ𝑧𝑓^′(𝑧)𝑓(𝑧)>0(𝑧∈𝕌). (4) Let 𝐾 denote the class of all functions 𝑓∈𝒜 that are convex. Further, 𝑓 is convex if and only if 𝑧𝑓^′ is starlike. Also we know that 𝐾⊂𝑆^∗⊂𝑆.

In 1959, Sakaguchi [2] introduced the class 𝑆^∗_𝑠 of functions starlike with respect to symmetric points, consisting of functions 𝑓∈𝑆 satisfying ℜ2𝑧𝑓^′(𝑧)𝑓(𝑧)()().−𝑓−𝑧>0𝑧∈𝕌 (5)

In 1977, Das and Singh [3] introduced the class 𝐾_𝑠 of functions convex with respect to symmetric points, which consists of functions 𝑓∈𝑆 satisfying ℜ2𝑧𝑓^′(𝑧)^′𝑓(𝑧)−𝑓(−𝑧)^′>0(𝑧∈𝕌). (6)

It is evident that 𝑓∈𝐾_𝑠 if and only if 𝑧𝑓^′∈𝑆^∗_𝑠.

In 2007, Wang and Jiang [4] introduced the following subclass.

Definition 1 (see [4]). Suppose that 0≤𝛼≤1 and 0<𝛽≤1. Let 𝑆(𝛼,𝛽) denote the class of functions 𝑓 in 𝒜 satisfying the following inequality: 𝑧𝑓^′(𝑧)𝑓(𝑧)−1<𝛽𝛼𝑧𝑓^′(𝑧)𝑓(𝑧)+1(𝑧∈𝕌). (7)
From [4], one knows that the above condition is equivalent to 𝑧𝑓^′(𝑧)𝑓(𝑧)≺1+𝛽𝑧(),1−𝛼𝛽𝑧𝑧∈𝕌 (8) which implies that 𝑆𝛼,𝛽⊂𝑆^∗⊂𝑆. (9)
If 𝛼=𝛽=1, then the class 𝑆(𝛼,𝛽) reduces to the class 𝑆^∗. In the similar way, one can easily get the following definitions.

Definition 2. Suppose that 0≤𝛼≤1 and 0<𝛽≤1. Let 𝐾(𝛼,𝛽) denote the class of functions 𝑓 in 𝒜 satisfying the following inequality: 𝑧𝑓^′(𝑧)^′𝑓^′(𝑧)𝛼−1<𝛽𝑧𝑓^′(𝑧)^′𝑓^′(𝑧)+1(𝑧∈𝕌). (10)
It is evident that the above condition is equivalent to 𝑧𝑓^′(𝑧)^′𝑓^′(𝑧)≺1+𝛽𝑧(),1−𝛼𝛽𝑧𝑧∈𝕌 (11) which implies that 𝐾𝛼,𝛽⊂𝐾⊂𝑆. (12)
If 𝛼=1 and 𝛽=1, then the class 𝐾(𝛼,𝛽) reduces to the class 𝐾.

Definition 3. Suppose that 0≤𝛼≤1 and 0<𝛽≤1. Let 𝑆^∗_𝑠(𝛼,𝛽) denote the class of functions 𝑓 in 𝒜 satisfying the following inequality: 2𝑧𝑓^′(𝑧)𝑓(𝑧)−𝑓(−𝑧)−1<𝛽2𝛼𝑧𝑓^′(𝑧)𝑓(𝑧)()).−𝑓−𝑧+1(𝑧∈𝕌 (13)
From [5], one knows that the above condition is equivalent to 2𝑧𝑓^′(𝑧)𝑓(𝑧)()≺−𝑓−𝑧1+𝛽𝑧().1−𝛼𝛽𝑧𝑧∈𝕌 (14)
The function class 𝑆^∗_𝑠(𝛼,𝛽) was introduced and investigated by Sudharsan et al. [6]. If 𝛼=1 and 𝛽=1, then the class 𝑆^∗_𝑠(𝛼,𝛽) reduces to the class 𝑆^∗_𝑠.

Definition 4. Suppose that 0≤𝛼≤1 and 0<𝛽≤1. Let 𝐾_𝑠(𝛼,𝛽) denote the class of functions 𝑓 in 𝒜 satisfying the following inequality: 2𝑧𝑓^′(𝑧)^′𝑓(𝑧)−𝑓(−𝑧)^′−1<𝛽2𝛼𝑧𝑓^′(𝑧)^′𝑓(𝑧)−𝑓(−𝑧)^′().+1𝑧∈𝕌 (15)
It is evident that the above condition is equivalent to 2𝑧𝑓^′(𝑧)^′𝑓(𝑧)−𝑓(−𝑧)^′≺1+𝛽𝑧().1−𝛼𝛽𝑧𝑧∈𝕌 (16)
If 𝛼=1 and 𝛽=1, then the class 𝐾_𝑠(𝛼,𝛽) reduces to the class 𝐾_𝑠.

In 1966, Pommerenke [7] stated the 𝑞th Hankel determinant for 𝑞≥1 and 𝑛≥1 as 𝐻_𝑞(𝑛)=𝑎_𝑛𝑎_𝑛+1⋯𝑎_{𝑛+𝑞−1}𝑎_𝑛+1𝑎_𝑛+2⋯𝑎_𝑛+𝑞𝑎⋯⋯⋯⋯_{𝑛+𝑞−1}𝑎_𝑛+𝑞⋯𝑎_{𝑛+2𝑞−2},𝑎₁=1. (17) This Hankel determinant is useful and has also been considered by several authors. The growth rate of Hankel determinant 𝐻_𝑞(𝑛) as 𝑛→∞ was investigated, respectively, when 𝑓 is a member of certain subclass of analytic functions, such as the class of p-valent functions [7, 8], the class of starlike functions [7], the class of univalent functions [9], the class of close-to-convex functions [10], the class of strong close-to-convex functions [11], a new class 𝑉_𝑘 [12], and a new class 𝑁_𝑘(𝜂,𝜌,𝛽) [13]. Similar to the above discussions, we can also refer to [14, 15]. Ehrenborg [16] studied the Hankel determinant of exponential polynomials. The Hankel transform of an integer sequence was defined and some of its properties were discussed by Layman [17]. Pommerenke [9] proved that the Hankel determinants of univalent function satisfy 𝐻_𝑞(𝑛)≤𝐾𝑛^{−(1/2+𝛽)𝑞+3/2}. (18) Later, |𝐻₂(𝑛)|≤𝐴𝑛^1/2 was also proved by Hayman [18]. One can easily observe that the Fekete and Szegö functional is 𝐻₂(1)=𝑎₃−𝑎²₂. For results related to the functional, see [19, 20]. Fekete and Szegö further generalized the estimate |𝑎₃−𝜇𝑎²₂|, where 𝜇 is real and 𝑓∈𝑆. For results related to the functional, see [21, 22]. In 2010, Hayami and Owa [21, 22] also generalized the estimate |𝑎_𝑛𝑎_𝑛+2−𝜇𝑎²_𝑛| for analytic function. Later, in 2012, Krishna and Ramreddy [23] also generalized the estimate |𝑎_𝑝+1𝑎_𝑝+3−𝜇𝑎²_𝑝+2| for p-valent analytic function; see also [24, 25].

For our discussion in this paper, we consider the second Hankel determinant in the case of 𝑞=2 and 𝑛=2, namely, 𝐻₂(2)=𝑎₂𝑎₃𝑎₃𝑎₄=𝑎₂𝑎₄−𝑎²₃. (19)

Janteng et al. [26] have considered the functional |𝐻₂(2)| and found a sharp bound, the subclass of 𝑆 denoted by 𝑅, defined as ℜ{𝑓^′(𝑧)}>0. In their work, they have shown that if 𝑓∈𝑅, then |𝐻₂(2)|≤4/9. These authors [27, 28] also studied the second Hankel determinant and sharp bound for the classes of starlike and convex functions, close-to-starlike and close-to-convex functions with respect to symmetric points denoted by 𝑆^∗, 𝐾, 𝑆^∗_𝑐, and 𝐾_𝑐 and have shown that |𝐻₂(2)|≤1, |𝐻₂(2)|≤1/8, |𝐻₂(2)|≤1, and |𝐻₂(2)|≤1/9, respectively.

Singh [29] established the second Hankel determinant and sharp bound for the classes of close-to-starlike and close-to-convex functions with respect to conjugate and symmetric conjugate points denoted by 𝑆^∗_𝑐, 𝑆^∗_𝑠𝑐, 𝐾_𝑐, and 𝐾_𝑠𝑐 and has shown that |𝐻₂(2)|≤1, |𝐻₂(2)|≤1, |𝐻₂(2)|≤1/8, and |𝐻₂(2)|≤1/9, respectively.

Mishra and Gochhayat [30] obtained the sharp bound to |𝐻₂(2)| for the functions in the class denoted by 𝑅_𝜆(𝛼,𝜌), (0≤𝜆<1,|𝛼|<𝜋/2,0≤𝜌≤1) and defined as ℜ{𝑒^𝑖𝛼(Ω^𝜆_𝑧𝑓(𝑧)/𝑧)}>𝜌cos𝛼, using the fractional differential operator denoted by Ω^𝜆_𝑧𝑓(𝑧) and defined by Owa and Srivastava [31]. These authors have shown that if 𝑓∈𝑅_𝜆(𝛼,𝜌), then |𝐻₂(2)|≤{((1−𝜌)²(2−𝜆)²(3−𝜆)²cos²𝛼)/9}.

Mohammed and Darus [32] have obtained a sharp upper bound to |𝐻₂(2)| for the functions in the class denoted by 𝑆_𝑚^𝜆,𝑛(𝛼,𝜎), (|𝛼|<𝜋/2,0≤𝜎<1) and defined as ℜ{𝑒^𝑖𝛼(Θ_𝑚^𝜆,𝑛𝑓(𝑧)/𝑧)}>𝜎cos𝛼. These authors have proved that if 𝑓∈𝑆_𝑚^𝜆,𝑛(𝛼,𝜎), then |𝐻₂(2)|≤{(4𝑚²(1−𝜎)²(1+𝑚)²cos²𝛼)/(3^2𝑛(𝜆+1)²(𝜆+2)²)}.

Similar to the above discussions in a new subclass of analytic function with different operators, we can also refer to [33, 34]. Singh [35] also obtained a sharp upper bound for the functional |𝐻₂(2)| for the function 𝑓∈𝑀(𝛼), where 𝑀(𝛼)=𝑓∈𝒜∶ℜ𝑧𝑓^′(𝑧)+𝛼𝑧²𝑓^′′(𝑧)(1−𝛼)𝑓(𝑧)+𝛼𝑧𝑓^′(𝑧),>0,0≤𝛼≤1,𝑧∈𝕌 (20) and showed that if 𝑓∈𝑀(𝛼), then |𝐻₂(2)|≤1/((1+𝛼)(1+3𝛼)).

Mehrok and Singh [36] have obtained a sharp upper bound to |𝐻₂(2)| for the function in the classes denoted by 𝑀^𝛼 and 𝐶_𝑠^∗(𝛼) and defined as, respectively, 𝑀^𝛼=𝑓∈𝒜∶ℜ𝑧𝑓^′(𝑧)𝑓(𝑧)^1−𝛼𝑧𝑓^′(𝑧)^′𝑓^′(𝑧)^𝛼,𝐶>0,0≤𝛼≤1,𝑧∈𝕌_{𝑠∗(𝛼)}=𝑓∈𝒜∶ℜ2𝑧𝑓^′(𝑧)𝑓(𝑧)−𝑓(−𝑧)^1−𝛼×2𝑧𝑓^′(𝑧)^′𝑓(𝑧)−𝑓(−𝑧)^′𝛼.>0,0≤𝛼≤1,𝑧∈𝕌 (21) In their work, they proved that if 𝑓∈𝑀^𝛼, then 𝐻₂(2≤1)(1+2𝛼)²×𝛼11+36𝛼+38𝛼²+12𝛼³−𝛼⁴×(1+3𝛼)−4+263𝛼+603𝛼²+253𝛼³+37𝛼⁴×(1+𝛼)⁴⁻¹+1, (22) and if 𝑓∈𝐶_𝑠^∗(𝛼), then |𝐻₂(2)|≤1/(1+2𝛼)².

Shanmugam et al. [37] established the sharp upper bound of the second Hankel determinant for the classes of 𝑆^∗_𝛼 and 𝐶_𝛼, defined as, respectively, 𝑆^∗_𝛼=𝑓∈𝒜∶ℜ𝑧𝑓^′(𝑧)𝑓(𝑧)𝑧+𝛼²𝑓^′′(𝑧)𝑓(𝑧),𝐶>0,𝑧∈𝕌_𝛼=𝑓∈𝒜∶ℜ𝑧𝑓^′(𝑧)+𝛼𝑧²𝑓^′′𝑧^′′𝑓^′(𝑧)>0,𝑧∈𝕌. (23) These authors proved that if 𝑓∈𝑆^∗_𝛼, then |𝐻₂(2)|≤1/(1+3𝛼)² and if 𝑓∈𝐶_𝛼, then 𝐻₂(2≤1)144280𝛼³+340𝛼²+138𝛼+18(1+2𝛼)²(1+3𝛼)²().1+4𝛼 (24)

Krishna and Ramreddy [38] obtained a sharp upper bound to the nonlinear functional |𝐻₂(2)| for a new subclass of analytic functions 𝑄(𝛼,𝛽,𝛾), (𝛼,𝛽>0,0≤𝛾<𝛼+𝛽≤1), defined by 𝑄=𝛼𝑓(𝑧)𝛼,𝛽,𝛾𝑓∈𝒜∶ℜ𝑧+𝛽𝑓^′(𝑧)≥𝛾,𝑧∈𝕌. (25) These authors proved that if 𝑓∈𝑄(𝛼,𝛽,𝛾), then |𝐻₂(2)|≤[4(𝛼+𝛽−𝛾)²/(𝛼+3𝛽)²].

Similar to the above discussions defined as different classes of analytic functions, we can also refer to [39–49]. Raza and Malik [50] studied the third Hankel determinant 𝐻₃(1) of analytic functions related with lemniscate of Bernoulli; see also [51].

Motivated by the above-mentioned results obtained by different authors in this direction, in this present investigation, we determine the upper bounds of the second Hankel determinant 𝐻₂(2) for functions belonging to these classes 𝑆(𝛼,𝛽), 𝐾(𝛼,𝛽), 𝑆^∗_𝑠(𝛼,𝛽), and 𝐾_𝑠(𝛼,𝛽).

2. Preliminary Results

In order to prove our main results, we need the following lemmas.

Lemma 5 (see [52]). If the function 𝑝∈𝒫 is given by the power series (2), then |𝑐_𝑘|≤2(𝑘=1,2,…).

Lemma 6 (see [53, 54]). If the function 𝑝∈𝒫 is given by the power series (2), then 2𝑐₂=𝑐²₁+4−𝑐²₁𝑥 (26) for some 𝑥 with |𝑥|≤1 and 4𝑐₃=𝑐³₁+2𝑐₁4−𝑐²₁𝑥−𝑐₁4−𝑐²₁𝑥²+24−𝑐²₁1−|𝑥|²𝑧 (27) for some 𝑧 with |𝑧|≤1.

3. Main Results

Theorem 7. Let 0≤𝛼≤1 and 0<𝛽≤1. Suppose that the function 𝑓 given by (1) is in the class 𝑆(𝛼,𝛽). Then 𝑎₂𝑎₄−𝑎²₃≤14𝛽²(1+𝛼)². (28) The result is sharp, with the extremal function 𝑓₁(𝑧)=𝑧1−𝛼𝛽𝑧^{2−(1+𝛼)/2𝛼},0<𝛼≤1,𝑧𝑒^𝛽𝑧2/2,𝛼=0. (29)

Proof. Since 𝑓∈𝑆(𝛼,𝛽), it follows from (8) that there exists a Schwarz function 𝜔, which is analytic in 𝕌 with 𝜔(0)=0 and |𝜔(𝑧)|<1 in 𝕌, such that 𝑧𝑓^′(𝑧)𝑓(𝑧)=𝜙(𝜔(𝑧))(𝑧∈𝕌), (30) where 𝜙(𝑧)=1+𝛽𝑧()1−𝛼𝛽𝑧=1+𝛽1+𝛼𝑧+𝛼𝛽²(1+𝛼)𝑧²+𝛼²𝛽³(1+𝛼)𝑧³+⋯. (31) Define the function 𝑝 by 𝑝(𝑧)=1+𝜔(𝑧)1−𝜔(𝑧)=1+𝑐₁𝑧+𝑐₂𝑧²+⋯. (32) From (3), we get 𝑝∈𝒫 and 𝜔(𝑧)=𝑝(𝑧)−1𝑝(𝑧)=1+12𝑐₁1𝑧+2𝑐₂−12𝑐²₁𝑧²+12𝑐₃−𝑐₁𝑐₂+14𝑐³₁𝑧³+⋯. (33) In view of (30), (31), and (33), we have 𝑧𝑓^′(𝑧)𝑓(𝑧)(𝜔(𝑧1=𝜙))=𝜙2𝑐₁1𝑧+2𝑐₂−12𝑐²₁𝑧²+12𝑐₃−𝑐₁𝑐₂+14𝑐³₁𝑧³1+⋯=1+2𝛽(1+𝛼)𝑐₁𝑧+12𝛽(1+𝛼)𝑐₂−12𝑐²₁+14𝛼𝛽²(1+𝛼)𝑐²₁𝑧²+12𝛽(1+𝛼)𝑐₃−𝑐₁𝑐₂+14𝑐³₁+12𝛼𝛽²(1+𝛼)𝑐₂−12𝑐²₁𝑐₁+18𝛼²𝛽³(1+𝛼)𝑐³₁𝑧³+⋯. (34) Similarly, 𝑧𝑓^′(𝑧)𝑓(𝑧)=1+𝑎₂𝑧+2𝑎₃−𝑎²₂𝑧²+3𝑎₄−3𝑎₂𝑎₃+𝑎²₂𝑧³+⋯. (35)
Comparing the coefficients of 𝑧, 𝑧², and 𝑧³ in (34) and (35), we obtain 𝑎₂=12𝛽(1+𝛼)𝑐₁,𝑎₃=18𝛽(1+𝛼)2𝑐₂+𝛽+2𝛼𝛽−1𝑐²₁,𝑎₄=18𝛽()×11+𝛼3−127𝛽−65𝛼𝛽+6𝛼𝛽²+𝛼²𝛽²+16𝛽²𝑐³₁−12𝛽()11+𝛼3−147𝛽−𝑐12𝛼𝛽₁𝑐₂+16𝛽(1+𝛼)𝑐₃. (36) Thus we have 𝑎₂𝑎₄−𝑎²₃1=−𝛽192²(1+𝛼)²×2𝛼𝛽²+2𝛼𝛽+𝛽²−1𝑐⁴₁−4𝛼𝛽−1𝑐²₁𝑐₂−16𝑐₁𝑐₃+12𝑐²₂,𝑎₂𝑎₄−𝑎²₃=1𝛽192²(1+𝛼)²×2𝛼𝛽²+2𝛼𝛽+𝛽²−1𝑐⁴₁−4𝛼𝛽−1𝑐²₁𝑐₂−16𝑐₁𝑐₃+12𝑐²₂. (37)(38)
Since the functions 𝑝(𝑧) and 𝑝(𝑒^𝑖𝜃𝑧)  (𝜃∈ℝ) are members of the class 𝒫 simultaneously, we assume without loss of generality that 𝑐₁>0. For convenience of notation, we take 𝑐₁=𝑐(𝑐∈[0,2]). By substituting the values of 𝑐₂ and 𝑐₃, respectively, from (26) and (27) in (38), we have 𝑎₂𝑎₄−𝑎²₃=1𝛽192²(1+𝛼)²×(2𝛼+1)𝛽²𝑐⁴−2𝛼𝛽𝑐²4−𝑐²𝑥+12+𝑐²4−𝑐²𝑥²−8𝑐4−𝑐²1−|𝑥|²𝑧. (39)
Using the triangle inequality and |𝑧|≤1, we have 𝑎₂𝑎₄−𝑎²₃≤1𝛽192²(1+𝛼)²×(2𝛼+1)𝛽²𝑐⁴+2𝛼𝛽𝑐²4−𝑐²|𝑥|+12+𝑐²4−𝑐²|𝑥|²+8𝑐4−𝑐²1−|𝑥|²=1𝛽192²(1+𝛼)²×8𝑐4−𝑐²+(2𝛼+1)𝛽²𝑐⁴+2𝛼𝛽𝑐²4−𝑐²|𝑥|+(𝑐−2)(𝑐−6)4−𝑐²|𝑥|²=𝐹𝑐,𝜇,𝑠𝑎𝑦, (40) where 𝜇=|𝑥|≤1.
We next maximize the function 𝐹(𝑐,𝜇) on the closed square [0,2]×[0,1]. Differentiating 𝐹(𝑐,𝜇) in (40) partially with respect to 𝜇, we get 𝜕𝐹𝑐,𝜇=1𝜕𝜇𝛽96²(1+𝛼)²×𝛼𝛽𝑐²4−𝑐²+(𝑐−2)(𝑐−6)4−𝑐²𝜇. (41) For 0<𝜇<1 and for any fixed 𝑐 with 0<𝑐<2, from (41), we observe that 𝜕𝐹(𝑐,𝜇)/𝜕𝜇>0. Consequently, 𝐹(𝑐,𝜇) is an increasing function of 𝜇 and hence it cannot have a maximum value at any point in the interior of the closed square [0,2]×[0,1]. Moreover, for fixed 𝑐∈[0,2], we have max_0≤𝜇≤1𝐹𝑐,𝜇=𝐹(𝑐,1)=𝐺(𝑐)𝑠𝑎𝑦. (42) From the relations (40) and (42), upon simplification, we obtain 𝐺(𝑐)()=1=𝐹𝑐,1𝛽192²(1+𝛼)²×2𝛼𝛽+𝛽+1𝛽−1𝑐⁴+8𝛼𝛽−1𝑐²+48. (43)
Next, since 𝐺^′(𝑐)=1𝛽48²(1+𝛼)²𝑐×2𝛼𝛽+𝛽+1𝛽−1𝑐²+4𝛼𝛽−1, (44) we get that 𝐺^′(𝑐)≤0 for 0<𝑐≤2 and 𝐺(𝑐) has real critical point at 𝑐=0. Therefore, the maximum of 𝐺(𝑐) occurs at 𝑐=0. Thus, the upper bound of 𝐹(𝑐,𝜇) corresponds to 𝜇=1 and 𝑐=0. Hence, 𝑎₂𝑎₄−𝑎²₃≤14𝛽²(1+𝛼)². (45)
Equality holds for the function 𝑓₁(𝑧)=𝑧1−𝛼𝛽𝑧^{2−(1+𝛼)/2𝛼},0<𝛼≤1,𝑧𝑒^𝛽𝑧2/2,𝛼=0. (46) By calculating, we have 𝑧𝑓^′₁(𝑧)𝑓₁(𝑧)=1+𝛽𝑧²1−𝛼𝛽𝑧²≺1+𝛽𝑧1−𝛼𝛽𝑧 (47) and 𝑎₂=0, 𝑎₃=(1/2)𝛽(1+𝛼), and 𝑎₄=0. So 𝑓₁(𝑧)∈𝑆(𝛼,𝛽) and equality holds. This shows that the result is sharp, and the proof of Theorem 7 is complete.

Setting 𝛼=𝛽=1 in Theorem 7, we obtain the following result due to Janteng et al. [27].

Corollary 8. If 𝑓(𝑧)∈𝑆^∗, then 𝑎₂𝑎₄−𝑎²₃≤1. (48) The result is sharp, with the extremal function 𝑓₂(𝑧)=𝑧1−𝑧². (49)

By using the similar method as in the proof of Theorem 7, one can similarly prove Theorem 9.

Theorem 9. Let 0≤𝛼≤1 and 0<𝛽≤1. Suppose that the function 𝑓 given by (1) is in the class 𝐾(𝛼,𝛽). Then𝑎₂𝑎₄−𝑎²₃≤^{1𝛽362()1+𝛼2}₁^{,5𝛼𝛽+𝛽−2≤0,}_{𝛽5762()1+𝛼25𝛼𝛽+𝛽−2}²_{()2+𝛽5𝛼+1−𝛽2()1−𝛼)(2𝛼+1+16,5𝛼𝛽+𝛽−2>0.} (50)The results are sharp, with the extremal function 𝑓₃(𝑧)=^𝑧01−𝛼𝛽𝜇^{2−(1+𝛼)/2𝛼}𝑑𝜇,0<𝛼≤1,_𝑧0𝑒^𝛽𝜇2/2𝑑𝜇,𝛼=0 (51) for the case 5𝛼𝛽+𝛽−2≤0, and there is no extremal function for the case 5𝛼𝛽+𝛽−2>0.

Setting 𝛼=𝛽=1 in Theorem 9, one obtains the following result due to Janteng et al. [27].

Corollary 10. If 𝑓(𝑧)∈𝐾, then 𝑎₂𝑎₄−𝑎²₃≤18. (52) The result is sharp.

Theorem 11. Let 0≤𝛼≤1 and 0<𝛽≤1. Suppose that the function 𝑓 given by (1) is in the class 𝑆^∗_𝑠(𝛼,𝛽). Then 𝑎₂𝑎₄−𝑎²₃≤14𝛽²(1+𝛼)². (53) The result is sharp, with the extremal function 𝑓₄(𝑧)=^𝑧01−𝛼𝛽𝜇^{2−(1+𝛼)/2𝛼}×1+𝛽𝜇²1−𝛼𝛽𝜇²𝑑𝜇,0<𝛼≤1,_𝑧0𝑒^𝛽𝜇2/21+𝛽𝜇²𝑑𝜇,𝛼=0. (54)

Proof. Since 𝑓∈𝑆^∗_𝑠(𝛼,𝛽), it follows from (14) that there exists a Schwarz function 𝜔, which is analytic in 𝕌 with 𝜔(0)=0 and |𝜔(𝑧)|<1 in 𝕌, such that 2𝑧𝑓^′(𝑧)𝑓(𝑧)()(𝜔(𝑧),−𝑓−𝑧=𝜙))(𝑧∈𝕌 (55) where 𝜙 was defined by (31).
In view of (31), (33), and (55), we have 2𝑧𝑓^′(𝑧)𝑓(𝑧)()(𝜔(𝑧1−𝑓−𝑧=𝜙))=𝜙2𝑐₁1𝑧+2𝑐₂−𝑐²₁2𝑧²+12𝑐₃−𝑐₁𝑐₂+𝑐³₁4𝑧³1+⋯=1+2𝛽(1+𝛼)𝑐₁𝑧+12𝛽(1+𝛼)𝑐₂−12𝑐²₁+14𝛼𝛽²(1+𝛼)𝑐²₁𝑧²+12𝛽(1+𝛼)𝑐₃−𝑐₁𝑐₂+14𝑐³₁+12𝛼𝛽²(1+𝛼)𝑐₂−12𝑐²₁𝑐₁+18𝛼²𝛽³(1+𝛼)𝑐³₁𝑧³+⋯. (56) Similarly, 2𝑧𝑓^′(𝑧)𝑓(𝑧)−𝑓(−𝑧)=2𝑎₂𝑧+2𝑎₃𝑧²+22𝑎₄−𝑎₂𝑎₃𝑧³+⋯. (57) Comparing the coefficients of 𝑧, 𝑧², and 𝑧³ in (56) and (57), we obtain 𝑎₂=14𝛽(1+𝛼)𝑐₁,𝑎₃=14𝛽(1+𝛼)𝛼𝛽−1𝑐²₁+2𝑐₂,𝑎₄=1𝛽()×641+𝛼2−4𝛼𝛽+3𝛼²𝛽²+𝛼𝛽²𝑐³₁+1𝛽()𝑐321+𝛼5𝛼𝛽+𝛽−4₁𝑐₂+18𝛽(1+𝛼)𝑐₃. (58) Thus we have 𝑎₂𝑎₄−𝑎²₃1=−𝛽256²(1+𝛼)²×𝛼²𝛽²−𝛼𝛽²−4𝛼𝛽+2𝑐⁴₁+6𝛼𝛽−2𝛽−8𝑐²₁𝑐₂−8𝑐₁𝑐₃+16𝑐²₂,𝑎₂𝑎₄−𝑎²₃=1𝛽256²(1+𝛼)²×𝛼²𝛽²−𝛼𝛽²−4𝛼𝛽+2𝑐⁴₁+6𝛼𝛽−2𝛽−8𝑐²₁𝑐₂−8𝑐₁𝑐₃+16𝑐²₂. (59)(60) Since the functions 𝑝(𝑧) and 𝑝(𝑒^𝑖𝜃𝑧)  (𝜃∈ℝ) are members of the class 𝒫 simultaneously, we assume without loss of generality that 𝑐₁>0. For convenience of notation, we take 𝑐₁=𝑐(𝑐∈[0,2]). By substituting the values of 𝑐₂ and 𝑐₃, respectively, from (26) and (27) in (60), we have 𝑎₂𝑎₄−𝑎²₃=1𝛽256²(1+𝛼)²×𝛼²𝛽²−𝛼𝛽²−𝛼𝛽−𝛽𝑐⁴+3𝛼𝛽−𝛽+4𝑐²4−𝑐²𝑥+24−𝑐²×8−𝑐²𝑥²−4𝑐4−𝑐²1−|𝑥|²𝑧. (61) Using the triangle inequality and |𝑧|<1, we have 𝑎₂𝑎₄−𝑎²₃≤1𝛽256²(1+𝛼)²×𝛽+𝛼𝛽+𝛼𝛽²−𝛼²𝛽²𝑐⁴+3𝛼𝛽−𝛽+4𝑐²4−𝑐²|𝑥|+24−𝑐²×8−𝑐²|𝑥|²+4𝑐4−𝑐²1−|𝑥|²=1𝛽256²(1+𝛼)²×𝛽+𝛼𝛽+𝛼𝛽²−𝛼²𝛽²𝑐⁴+4𝑐4−𝑐²+4+3𝛼𝛽−𝛽𝑐²4−𝑐²𝑥|(|+22−𝑐)(4+𝑐)4−𝑐²|𝑥|²=𝐹𝑐,𝜇,𝑠𝑎𝑦, (62) where 𝜇=|𝑥|≤1.
We next maximize the function 𝐹(𝑐,𝜇) on the closed square [0,2]×[0,1]. Differentiating 𝐹(𝑐,𝜇) in (62) partially with respect to 𝜇, we get 𝜕𝐹𝑐,𝜇=1𝜕𝜇𝛽256²(1+𝛼)²×4+3𝛼𝛽−𝛽𝑐²4−𝑐²+4(2−𝑐)(4+𝑐)4−𝑐²𝜇. (63) For 0<𝜇<1 and for any fixed 𝑐 with 0<𝑐<2, from (63), we observe that 𝜕𝐹(𝑐,𝜇)/𝜕𝜇>0. Consequently, 𝐹(𝑐,𝜇) is an increasing function of 𝜇 and hence it cannot have a maximum value at any point in the interior of the closed square [0,2]×[0,1]. Moreover, for fixed 𝑐∈[0,2], we have max_0≤𝜇≤1𝐹𝑐,𝜇=𝐹(𝑐,1)=𝐺(𝑐)𝑠𝑎𝑦. (64)
From the relations (62) and (64), upon simplification, we obtain 𝐺(𝑐)()=1=𝐹𝑐,1𝛽256²(1+𝛼)²×2𝛽−2𝛼𝛽+𝛼𝛽²−𝛼²𝛽²−2𝑐⁴+43𝛼𝛽−𝛽−2𝑐²+64. (65) Next, since 𝐺^′(𝑐)=1𝛽64²(1+𝛼)²𝑐×2𝛽−2𝛼𝛽+𝛼𝛽²−𝛼²𝛽²−2𝑐²+23𝛼𝛽−𝛽−2, (66) we get that 𝐺^′(𝑐)≤0 for 0<𝑐≤2 and 𝐺(𝑐) has real critical point at 𝑐=0. Therefore, the maximum of 𝐺(𝑐) occurs at 𝑐=0. Thus, the upper bound of 𝐹(𝑐,𝜇) corresponds to 𝜇=1 and 𝑐=0. Hence, 𝑎₂𝑎₄−𝑎²₃≤14𝛽²(1+𝛼)². (67)
Equality holds for the function 𝑓₄(𝑧)=^𝑧01−𝛼𝛽𝜇^{2−(1+𝛼)/2𝛼}×1+𝛽𝜇²1−𝛼𝛽𝜇²𝑑𝜇,0<𝛼≤1,_𝑧0𝑒^𝛽𝜇2/21+𝛽𝜇²𝑑𝜇,𝛼=0. (68) By calculating, we have 2𝑧𝑓^′₄(𝑧)𝑓₄(𝑧)−𝑓₄()=−𝑧1+𝛽𝑧²1−𝛼𝛽𝑧²≺1+𝛽𝑧1−𝛼𝛽𝑧 (69) and 𝑎₂=0, 𝑎₃=−(1/2)𝛽(1+𝛼), and 𝑎₄=0. So 𝑓₄(𝑧)∈𝑆(𝛼,𝛽) and equality holds. This shows that the result is sharp, and the proof of Theorem 11 is complete.

Setting 𝛼=𝛽=1 in Theorem 11, we obtain the following result due to Janteng et al. [28].

Corollary 12. If 𝑓(𝑧)∈𝑆^∗_𝑠, then 𝑎₂𝑎₄−𝑎²₃≤1. (70) The result is sharp, with the extremal function 𝑓₅(𝑧)=^𝑧₀1+𝜇²1−𝜇²²𝑑𝜇. (71)

By using the similar method as in the proof of Theorem 11, one can similarly prove Theorem 13.

Theorem 13. Let 0≤𝛼≤1 and 0<𝛽≤1. Suppose that the function 𝑓(𝑧) given by (1) is in the class 𝐾_𝑠(𝛼,𝛽). Then 𝑎₂𝑎₄−𝑎²₃≤1𝛽36²(1+𝛼)². (72) The result is sharp, with the extremal function 𝑓₆(𝑧)=^𝑧01𝜔^𝜔₀22−𝛼𝛽𝜇^{2(1+𝛼)/2𝛼}×2+𝛽𝜇²2−𝛼𝛽𝜇²𝑑𝜇𝑑𝜔,0<𝛼≤1,_𝑧01𝜔^𝜔₀𝑒^𝛽𝜇2/21+𝛽𝜇²2𝑑𝜇𝑑𝜔,𝛼=0. (73)

Setting 𝛼=𝛽=1 in Theorem 13, one obtains the following result due to Janteng et al. [28].

Corollary 14. If 𝑓(𝑧)∈𝐾_𝑠, then 𝑎₂𝑎₄−𝑎²₃≤19. (74) The result is sharp, with the extremal function 𝑓₇(𝑧)=2^𝑧₀1𝜔^𝜔₀2+𝜇²2−𝜇²²𝑑𝜇𝑑𝜔. (75)

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was financially supported by Foundation for Distinguished Young Talents in Higher Education of Guangdong China (no. 2013LYM0093) and Training plan for the Outstanding Young Teachers in Higher Education of Guangdong (no. Yq 2013159). The authors are also grateful to the anonymous referees and Professor V. Ravichandran for making many valuable suggestions that improved the quality and the readability of this paper.