Abstract
In the present investigation, we introduce the subclasses $\varLambda_{\Sigma}^{m}(\eta,\leftthreetimes,\phi)$ and $\varLambda_{\Sigma}^{m}(\eta,\leftthreetimes,\delta)$ of \textit{m}-fold symmetric bi-univalent function class $\Sigma_m$, which are associated with the pseudo-starlike functions and defined in the open unit disk $\mathbb{U}$. Moreover, we obtain estimates on the initial coefficients $|b_{m+1}|$ and $|b_{2m+1}|$ for the functions belong to these subclasses and identified correlations with some of the earlier known classes.
Highlights
Let A = {f : U → C : f is analytic in U, f (0) = 0 = f (0) − 1} be the class of functions of the form ∞f (z) = z + akzk (1.1)k=2 and S be the subclass of A consisting of all functions f univalent in U
With reference to the -pseudo-starlike function class defined by Babalola [3] and the work of Joshi and Yadav [10], we obtain estimates on the initial coefficients |bm+1| and |b2m+1| for functions belong to the new subclasses ΛmΣ (η, φ) and ΛmΣ (η, δ) of the function class Σm
For m = 1 symmetric bi-univalent function, Theorems 2.2 and 3.2 gives Corollaries 3.7 and 3.8, respectively, which were investigated by Joshi and Yadav [10]
Summary
Let A = {f : U → C : f is analytic in U, f (0) = 0 = f (0) − 1} be the class of functions of the form [1], [5], [7], [9], [14], [17], [18], [20], [21], [22] etc.) obtained coefficient estimates for the functions in several subclasses of Σ.
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