Let be the class of univalent analytic functions of the form

defined in the unit disk := {z ∈ ℂ : |z| < 1}. It is well-known that the coefficient of the functions in the class satisfy |a_n| ≤ n. This result was put before, as a conjecture, by Bieberbach in 1916, and it took around 68 year to prove and was finally affirmatively settled by de Branges. In those 68 years many researchers tried to prove or disprove it which lead to explore many subclasses of the class . The class of starlike functions is a collection of functions f ∈ for which Re(zf′(z)/f(z)) > 0 for all z ∈ . However, the class of convex functions is a collection of all those functions f ∈ for which Re(1 + zf″ (z)/f′(z)) > 0 for all z ∈ . These subclasses are among the most studied subclasses of . In 1997, Silverman [18] investigated a class of normalised analytic functions involving an expression of the quotient of the analytic representations of convex and starlike functions. For 0 < μ ≤ 1, he defined the class as follows:

and proved that the function in the class are starlike of order z/(1+1+8b). He also investigated many sufficient conditions for functions to be starlike and convex of positive order. Further, this result was improved by Obradovič and Tuneski [12]. In 2003, Tuneski [19] investigated the condition for functions in the class to be Janowski starlike. For further related results reader can refer [1, 7, 11] and the references cited therein.

Recall that for analytic functions f and g, we say that f is subordinate to g, denoted by f ≺ g, if there is a Schwarz function w with |w(z)| ≤ |z| such that f(z) = g(w(z)). Further, if g is univalent, then f ≺ g if and only if f(0) = g(0) and . In view of this definition, we can write

The sharp bound on the functional ∣a2a4-a32∣ for starlike and convex functions were obtained by Janteng [5]. He proved that for starlike and convex functions, this quantity is bounded above by 1 and 1/8, respectively. This functional is related to the Hankel determinants. Recall that for given natural numbers n, q, the Hankel determinant H_q,n(f) of a function is defined by means of the following determinant

with a₁ = 1. The quantity H2,1(f)=a3-a22 is the well-known Fekete-Szegö functional. The second Hankel determinant is given by the expression H2,2(f):=a2a4-a32. Further, the quantity H3,1(f):=a3(a2a4-a32)-a4(a4-a2a3)+a5(a3-a22) is called the third Hankel determinant. The Hankel determinant H_q,n(f) for the class of univalent functions was investigated by Pommerenke [14] and Hayman [4]. For a chronological development in this direction till 2013 reader may refer [8].

The results related to the Hankel determinants usually are derived by relating the functions in the class under consideration to the Carathéodory functions. For this purpose we recall the definition of this class. Let ℘ denote the class of Carathéodory [2, 3] functions of the form

The following results shall be used as tools:

Lemma 1.1

([9, 10, Libera and Zlotkiewicz]) If p ∈ ℘ has the form given by (1.2) with p₁ ≥ 0, then

and

for some x and y such that |x| ≤ 1 and |y| ≤ 1.

Lemma 1.2

([17, Ravichandran and Verma]) Let α̂, β̂, γ̂ and â satisfy the inequalities 0 < α̂ < 1, 0 < â < 1 and

If p ∈ ℘ has the form given by (1.2), then

The following result is due to Prokhorov and Szynal [15]. Since the result is lengthy, so we are quoting here some specific part of their result which we need in our further investigation.

Let ℬ be the class of analytic functions w(z)=∑n=1∞cnzn   (z∈D) and satisfying the condition |w(z)| < 1 for z ∈ . Consider a functional Ψ(w)= ∣c3+αc1c2+βc13∣ for w ∈ ℬ and α, β ∈ ℝ. Define the sets Ω₁,Ω₂,Ω₃,Ω₄ and Ω₄ by

and

Lemma 1.3

([15, Lemma 2, p. 128]) If w ∈ ℬ, then for any real numbers α and β the sharp estimate Ψ(w) ≤ Φ(α, β) holds, where

Lemma 1.4

([13, Ohno and Sugawa]) For any real real numbers a, b and c, let the quantity Y (a, b, c) be given by

where := {z ∈ ℂ : |z| ≤ 1}. If ac ≥ 0, then

Further, if ac < 0, then

where

The following theorem gives the sharp upper bound for the initial coefficients for functions in the class .

Theorem 2.1

Let f ∈ . Assume that μ₀ ≈ 0.335 be the smallest positive root of

Then the following sharp inequalities hold:

• (1) |a₂| ≤ μ,

• (2) ∣a3∣ ≤{μ4,0<μ≤1/4;μ2,1/4≤μ≤1,

• (3) ∣a4∣ ≤{μ9,0<μ≤1/3;μ3,1/3≤μ≤1,

• (4) ∣a5∣ ≤μ16 (0<μ≤μ0),

• (5) ∣a3-νa22∣ ≤μ4 max {1;4μ∣ν-1∣}, ν ∈ ℂ.

Theorem 2.2

Let f ∈ . Then the following sharp inequalities hold:

• (1) ∣a2a4-a32∣ ≤{μ216,0<μ≤16;μ2 (4+3μ)236(7+12μ),16<μ≤1.

• (2) ∣a2a3-a4∣ ≤μ9.

Corollary 2.3

Let f ∈ . Then the following holds:

Remark 2.4

It should be noted that all the estimates derived so far in this paper are sharp except the bound on the third Hankel determinant |H_{3, 1}(f)| mentioned in Corollary 2.3. Finding the sharp bound on the third Hankel determinants for many classes of analytic functions, including the class under consideration here, are still unsettled. However, here we have a partial solution for sharp bound on the third Hankel determinant under certain conditions.

It is known that a function is said to be n-fold symmetric if f(εz) = εf(z) holds for all z ∈ , where ε = e^2πi/n. The set of all n-fold symmetric functions is denoted by and functions in this class are of the form g_n(z) = z +a_n+1zⁿ⁺¹ + a_2n+1z²ⁿ⁺¹ + ⋯. In particular, any function g in the class has the form g₃(z) = z + b₄z⁴ + b₇z⁷ + ⋯. Thus it is clear that ∣H3,1(f)∣ = ∣b42∣. Since the functions in the class are starlike, it follows that f ∈ if and only if g3(z)=f(z3)3=z+(a2/3)z4+⋯∈Gμ. Thus b₄ = a₂/3 and ∣H3,1(f)∣ = ∣b42∣ = ∣a2∣2/9. Now from the first part of Theorem 2.1, we have |H_{3, 1}(f)| = |a₂|²/9 ≤ μ²/9. The result is sharp in case of the function f₀ defined by

Remark 2.5

A close observation on the results in Theorem 2.1 and the extremal functions reveals the following expected results, which have already been verified for n = 2, 3, 4 in Theorem 2.1. Further in the same theorem this conjecture is also verified for n = 5 for the range 0 < μ ≤ μ₀ ≈ 0.335.

Conjecture 2.6

Let f ∈ . Then, for any natural number n ≥ 5, the following sharp inequalities hold:

The extremal function, in both the cases 0 < μ ≤ (n − 1) ^−2/(n−2) and (n − 1) ^−2/(n−2) ≤ μ ≤ 1, is given by (2.2) with choice of the function w(z) = zⁿ⁻¹ and w(z) = z, respectively.