Successive coefficients for spirallike and related functions

Let $\mathcal{A}$ denote the set of all analytic functions $f$ on the unit disk normalised so that $f(0)=f'(0)-1=0$ and $\mathcal{S} $ denote the subclass of functions $f \in \mathcal{A}$ which are univalent in the unit disk.
The problem of estimating sharp bound for successive coefficients, namely, $\big | |a_{n+1}|-|a_n| \big | $, is an interesting necessary condition for a function to be in $\mathcal{S}$. This problem was first studied by Goluzin [1] with an idea to solve the Bieberbach conjecture. Hayman [3] proved in 1963 that
\begin{equation}\label{deq5} \big | |a_{n+1}|-|a_n| \big | \leq A, \quad n=1,2,3,\dots, \end{equation}
where $A\geq 1$ is an absolute constant, for $f \in \mathcal{S}$ with the form $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. It is still an open problem to find the minimal value of $A$ which works for all $f \in \mathcal{S}$, however, the best known bound as of now is $3.61$ which is due to Grinspan [2]. In $1978$, Leung [4] proved that $A=1$ for starlike functions implies the Bieberbach conjecture, so it is interesting to find the minimal value of $A$ for certain other subfamilies of univalent functions.
We are considering some subclasses of the class of univalent functions, mainly, the family $\mathcal{S}_{\gamma}(\alpha)$ of $\gamma$-spirallike functions of order $\alpha$ which is defined as

$$ \mathcal{S}_\gamma (\alpha) = \left \{f\in {\mathcal A}: \, {\rm Re} \left ( e^{-i\gamma}\frac{zf'(z)}{f(z)}\right )>\alpha \cos \gamma\, , z\in \mathbb{D}\right\}, $$
where $ \alpha \in [0,1)$ and $\gamma\in (-\pi/2, \pi/2)$. Note that $\mathcal{S}_0 (\alpha)=:\mathcal{S}^*{(\alpha)}$ is the usual class of starlike functions of order $\alpha$, and $\mathcal{S}^*(0)=\mathcal{S}^*$. A function $f \in \mathcal{A}$ is called convex of order $\alpha$ if and only if $zf'(z)$ belongs to $\mathcal{S}^*{(\alpha)}$ for some $\alpha \in [ 0,1)$. The class of all convex functions of order $\alpha$ is denoted by $\mathcal{C}(\alpha)$.
Our objective is to obtain results related to successive coefficients for the classes $\mathcal{S}^*{(\alpha)}$, $\mathcal{C}(\alpha)$, $\mathcal{S}_{\gamma}(\alpha)$ and other related classes of functions.