1. Introduction
Let A denote the class of functions of the form
(1)f(z)=z+∑k=2∞akzk
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
(2)p(z)=1+c1z+c2z2+⋯
for z∈U.

It is well known that the following correspondence between the class P and the class of Schwarz functions ω exists [1]:
(3)p∈P⟺p=1+ω1-ω.

Let S* denote the starlike subclass of S. It is well known that f∈S* if and only if
(4)R{zf′(z)f(z)}>0 (z∈U).
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 Ss* of functions starlike with respect to symmetric points, consisting of functions f∈S satisfying
(5)R{2zf′(z)f(z)-f(-z)}>0 (z∈U).

In 1977, Das and Singh [3] introduced the class Ks of functions convex with respect to symmetric points, which consists of functions f∈S satisfying
(6)R{2(zf′(z))′(f(z)-f(-z))′}>0 (z∈U).

It is evident that f∈Ks if and only if zf′∈Ss*.

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

Definition 1 (see [<xref ref-type="bibr" rid="B53">4</xref>]).
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:
(7)|zf′(z)f(z)-1|<β|αzf′(z)f(z)+1| (z∈U).

From [4], one knows that the above condition is equivalent to
(8)zf′(z)f(z)≺1+βz1-αβz (z∈U),
which implies that
(9)S(α,β)⊂S*⊂S.

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:
(10)|(zf′(z))′f′(z)-1|<β|α(zf′(z))′f′(z)+1| (z∈U).

It is evident that the above condition is equivalent to
(11)(zf′(z))′f′(z)≺1+βz1-αβz (z∈U),
which implies that
(12)K(α,β)⊂K⊂S.

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 Ss*(α,β) denote the class of functions f in A satisfying the following inequality:
(13)|2zf′(z)f(z)-f(-z)-1|<β|2αzf′(z)f(z)-f(-z)+1| (z∈U).

From [5], one knows that the above condition is equivalent to
(14)2zf′(z)f(z)-f(-z)≺1+βz1-αβz (z∈U).

The function class Ss*(α,β) was introduced and investigated by Sudharsan et al. [6]. If α=1 and β=1, then the class Ss*(α,β) reduces to the class Ss*.

Definition 4.
Suppose that 0≤α≤1 and 0<β≤1. Let Ks(α,β) denote the class of functions f in A satisfying the following inequality:
(15)|2(zf′(z))′(f(z)-f(-z))′-1|<β|2α(zf′(z))′(f(z)-f(-z))′+1|kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk(z∈U).

It is evident that the above condition is equivalent to
(16)2(zf′(z))′(f(z)-f(-z))′≺1+βz1-αβz (z∈U).

If α=1 and β=1, then the class Ks(α,β) reduces to the class Ks.

In 1966, Pommerenke [7] stated the qth Hankel determinant for q≥1 and n≥1 as
(17)Hq(n)=|anan+1⋯an+q-1an+1an+2⋯an+q⋯⋯⋯⋯an+q-1an+q⋯an+2q-2|, (a1=1).
This Hankel determinant is useful and has also been considered by several authors. The growth rate of Hankel determinant Hq(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 Vk [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
(18)|Hq(n)|≤Kn-(1/2+β)q+3/2.
Later, |H2(n)|≤An1/2 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 |anan+2-μan2| for analytic function. Later, in 2012, Krishna and Ramreddy [23] also generalized the estimate |ap+1ap+3-μap+22| 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,
(19)H2(2)=|a2a3a3a4|=a2a4-a32.

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, Sc*, and Kc 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 Sc*, Ssc*, Kc, and Ksc 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 Rλ(α,ρ), (0≤λ<1,|α|<π/2,0≤ρ≤1) and defined as R{eiα(Ωzλf(z)/z)}>ρcosα, using the fractional differential operator denoted by Ωzλf(z) and defined by Owa and Srivastava [31]. These authors have shown that if f∈Rλ(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 Smλ,n(α,σ), (|α|<π/2,0≤σ<1) and defined as R{eiα(Θmλ,nf(z)/z)}>σcosα. These authors have proved that if f∈Smλ,n(α,σ), then |H2(2)|≤{(4m2(1-σ)2(1+m)2cos2α)/(32n(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
(20)M(α)={∑f∈A:R[zf′(z)+αz2f′′(z)(1-α)f(z)+αzf′(z)]>0, 0≤α≤1,z∈U[zf′(z)+αz2f′′(z)(1-α)f(z)+αzf′(z)]∑},
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 Mα and Cs*(α) and defined as, respectively,
(21)Mα={∑f∈A:R[((zf′(z))′f′(z))α(zf′(z)f(z))1-α((zf′(z))′f′(z))α]>0, 0≤α≤1,z∈U∑[(zf′(z)f(z))1-α((zf′(z))′f′(z))α]},Cs*(α)={f∈A:R[(2(zf′(z))′(f(z)-f(-z))′)α(2zf′(z)f(z)-f(-z))1-α ×(2(zf′(z))′(f(z)-f(-z))′)α]>0, 0≤α≤1,z∈U[(2zf′(z)f(z)-f(-z))1-α}.
In their work, they proved that if f∈Mα, then
(22)|H2(2)| ≤1(1+2α)2 ×[(1+α)4)-1)α(11+36α+38α2+12α3-α4) ×((1+3α)(-4+263α+603α2+253α3+37α4) k ×(1+α)4)-1+1((α(11+36α+38α2+12α3-α4))],
and if f∈Cs*(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 Sα* and Cα, defined as, respectively,
(23)Sα*={∑f∈A:R[zf′(z)f(z)+αz2f′′(z)f(z)]>0,z∈U∑},Cα={∑f∈A:R[(zf′(z)+αz2f′′(z′))′f′(z)]>0,z∈U∑}.
These authors proved that if f∈Sα*, then |H2(2)|≤1/(1+3α)2 and if f∈Cα, then
(24)|H2(2)|≤1144|280α3+340α2+138α+18(1+2α)2(1+3α)2(1+4α)|.

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
(25)Q(α,β,γ)={f∈A:R[αf(z)z+βf′(z)]≥γ,z∈U}.
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), Ss*(α,β), and Ks(α,β).

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
(28)|a2a4-a32|≤14β2(1+α)2.
The result is sharp, with the extremal function
(29)f1(z)={z(1-αβz2)-(1+α)/2α,0<α≤1,zeβz2/2,α=0.

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
(30)zf′(z)f(z)=ϕ(ω(z)) (z∈U),
where
(31)ϕ(z)=1+βz1-αβz=1+β(1+α)z+αβ2(1+α)z2 +α2β3(1+α)z3+⋯.
Define the function p by
(32)p(z)=1+ω(z)1-ω(z)=1+c1z+c2z2+⋯.
From (3), we get p∈P and
(33)ω(z)=p(z)-1p(z)+1=12c1z+12(c2-12c12)z2 +12(c3-c1c2+14c13)z3+⋯.
In view of (30), (31), and (33), we have
(34)zf′(z)f(z)=ϕ(ω(z))=ϕ(12c1z+12(c2-12c12)z2 +12(c3-c1c2+14c13)z3+⋯)=1+12β(1+α)c1z +[12β(1+α)(c2-12c12)+14αβ2(1+α)c12]z2 +[12β(1+α)(c3-c1c2+14c13) +12αβ2(1+α)(c2-12c12)c1 +18α2β3(1+α)c13(c3-c1c2+14c13)]z3+⋯.
Similarly,
(35)zf′(z)f(z)=1+a2z+(2a3-a22)z2 +(3a4-3a2a3+a22)z3+⋯.

Comparing the coefficients of z, z2, and z3 in (34) and (35), we obtain
(36)a2=12β(1+α)c1,a3=18β(1+α)[2c2+(β+2αβ-1)c12],a4=18β(1+α) ×(13-12β-76αβ+56αβ2+α2β2+16β2)c13 -12β(1+α)(13-14β-712αβ)c1c2+16β(1+α)c3.
Thus we have
(37)a2a4-a32=-1192β2(1+α)2 ×[(2αβ2+2αβ+β2-1)c14-4(αβ-1)c12c2 -16c1c3+12c22],(38)|a2a4-a32|=1192β2(1+α)2 ×|(2αβ2+2αβ+β2-1)c14 k-4(αβ-1)c12c2-16c1c3+12c22|.

Since the functions p(z) and p(eiθz) (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
(39)|a2a4-a32|=1192β2(1+α)2 ×|(2α+1)β2c4-2αβc2(4-c2)x +(12+c2)(4-c2)x2 -8c(4-c2)(1-|x|2)z|.

Using the triangle inequality and |z|≤1, we have
(40)|a2a4-a32|≤1192β2(1+α)2 ×[(2α+1)β2c4+2αβc2(4-c2)|x| +(12+c2)(4-c2)|x|2 +8c(4-c2)(1-|x|2)]=1192β2(1+α)2 ×[8c(4-c2)+(2α+1)β2c4 +2αβc2(4-c2)|x| +(c-2)(c-6)(4-c2)|x|2]=F(c,μ), (say),
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
(41)∂F(c,μ)∂μ=196β2(1+α)2 ×[αβc2(4-c2)+(c-2)(c-6)(4-c2)μ].
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
(42)max0≤μ≤1F(c,μ)=F(c,1)=G(c) (say).
From the relations (40) and (42), upon simplification, we obtain
(43)G(c)=F(c,1)=1192β2(1+α)2 ×[(2αβ+β+1)(β-1)c4+8(αβ-1)c2+48].

Next, since
(44)G′(c)=148β2(1+α)2c ×[(2αβ+β+1)(β-1)c2+4(αβ-1)],
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,
(45)|a2a4-a32|≤14β2(1+α)2.

Equality holds for the function
(46)f1(z)={z(1-αβz2)-(1+α)/2α,0<α≤1,zeβz2/2,α=0.
By calculating, we have
(47)zf1′(z) f1(z)=1+βz21-αβz2≺1+βz1-αβz
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
(48)|a2a4-a32|≤1.
The result is sharp, with the extremal function
(49)f2(z)=z1-z2.

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). Then(50)|a2a4-a32|≤{136β2(1+α)2,5αβ+β-2≤0,1576β2(1+α)2[(5αβ+β-2)22+β(5α+1)-β2(1-α)(2α+1)+16],5αβ+β-2>0.The results are sharp, with the extremal function
(51)f3(z)={∫0z(1-αβμ2)-(1+α)/2αdμ,0<α≤1,∫0zeβμ2/2dμ,α=0
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
(52)|a2a4-a32|≤18.
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 Ss*(α,β). Then
(53)|a2a4-a32|≤14β2(1+α)2.
The result is sharp, with the extremal function
(54)f4(z)={∫0z(1-αβμ2)-(1+α)/2α ×(1+βμ21-αβμ2)dμ,0<α≤1,∫0zeβμ2/2(1+βμ2)dμ,α=0.

Proof.
Since f∈Ss*(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
(55)2zf′(z)f(z)-f(-z)=ϕ(ω(z)) (z∈U),
where ϕ was defined by (31).

In view of (31), (33), and (55), we have
(56)2zf′(z)f(z)-f(-z) =ϕ(ω(z)) =ϕ(12c1z+12(c2-c122)z2 +12(c3-c1c2+c134)z3+⋯) =1+12β(1+α)c1z +[12β(1+α)(c2-12c12)+14αβ2(1+α)c12]z2 +[12β(1+α)(c3-c1c2+14c13) +12αβ2(1+α)(c2-12c12)c1 +18α2β3(1+α)c13(c3-c1c2+14c13)]z3+⋯.
Similarly,
(57)2zf′(z)f(z)-f(-z)=2a2z+2a3z2+2(2a4-a2a3)z3+⋯.
Comparing the coefficients of z, z2, and z3 in (56) and (57), we obtain
(58)a2=14β(1+α)c1,a3=14β(1+α)[(αβ-1)c12+2c2],a4=164β(1+α) ×(2-4αβ+3α2β2+αβ2)c13 +132β(1+α)(5αβ+β-4)c1c2 +18β(1+α)c3.
Thus we have
(59)a2a4-a32=-1256β2(1+α)2 ×[(α2β2-αβ2-4αβ+2)c14 k +(6αβ-2β-8)c12c2-8c1c3+16c22],(60)|a2a4-a32|=1256β2(1+α)2 ×|(α2β2-αβ2-4αβ+2)c14 k +(6αβ-2β-8)c12c2-8c1c3+16c22|.
Since the functions p(z) and p(eiθz) (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
(61)|a2a4-a32|=1256β2(1+α)2 ×|(α2β2-αβ2-αβ-β)c4 +(3αβ-β+4)c2(4-c2)x+2(4-c2) ×(8-c2)x2-4c(4-c2)(1-|x|2)z|.
Using the triangle inequality and |z|<1, we have
(62)|a2a4-a32|≤1256β2(1+α)2 ×[(β+αβ+αβ2-α2β2)c4 +(3αβ-β+4)c2(4-c2)|x|+2(4-c2) ×(8-c2)|x|2+4c(4-c2)(1-|x|2)]=1256β2(1+α)2 ×[(β+αβ+αβ2-α2β2)c4+4c(4-c2) +(4+3αβ-β)c2(4-c2)|x| +2(2-c)(4+c)(4-c2)|x|2]=F(c,μ), (say),
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
(63)∂F(c,μ)∂μ=1256β2(1+α)2 ×[(4+3αβ-β)c2(4-c2) k +4(2-c)(4+c)(4-c2)μ].
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
(64)max0≤μ≤1F(c,μ)=F(c,1)=G(c) (say).

From the relations (62) and (64), upon simplification, we obtain
(65)G(c)=F(c,1)=1256β2(1+α)2 ×[(2β-2αβ+αβ2-α2β2-2)c4 +4(3αβ-β-2)c2+64].
Next, since
(66)G′(c)=164β2(1+α)2c ×[(2β-2αβ+αβ2-α2β2-2)c2 +2(3αβ-β-2)(2β-2αβ+αβ2-α2β2-2)],
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,
(67)|a2a4-a32|≤14β2(1+α)2.

Equality holds for the function
(68)f4(z)={∫0z(1-αβμ2)-(1+α)/2α ×(1+βμ21-αβμ2)dμ,0<α≤1,∫0zeβμ2/2(1+βμ2)dμ,α=0.
By calculating, we have
(69)2zf4′(z)f4(z)-f4(-z)=1+βz21-αβz2≺1+βz1-αβz
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)∈Ss*, then
(70)|a2a4-a32|≤1.
The result is sharp, with the extremal function
(71)f5(z)=∫0z1+μ2(1-μ2)2dμ.

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 Ks(α,β). Then
(72)|a2a4-a32|≤136β2(1+α)2.
The result is sharp, with the extremal function
(73)f6(z)={∫0z1ω{∫0ω(22-αβμ2)(1+α)/2α ×(2+βμ22-αβμ2)dμ}dω,0<α≤1,∫0z1ω{∫0ωeβμ2/2(1+βμ22)dμ}dω,α=0.

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

Corollary 14.
If f(z)∈Ks, then
(74)|a2a4-a32|≤19.
The result is sharp, with the extremal function
(75)f7(z)=2∫0z1ω{∫0ω2+μ2(2-μ2)2dμ}dω.