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In this paper, we introduce and investigate a subclass of analytic and bi-univalent functions which both f⁢(z){f(z)}and f-1⁢(z){f^{-1}(z)}are m-fold symmetric on the open unit disk. Furthermore, we find upper bounds for the initial coefficients |am+1|{|a_{m+1}|}and |a2⁢m+1|{|a_{2m+1}|}for functions in this subclass. The results presented in this paper generalize and improve some recent works

In this paper, we introduce and investigate an interesting subclass of meromorphic bi-univalent functions defined on Δ = {z ∈ ℂ: 1 < |z| < ∞}. For functions belonging to this class, estimates on the initial coefficients are obtained. The results presented in this paper generalize and improve some recent works.

AbstractLet p(z) be a polynomial of degree n,and Dαp(z) be its polar derivative of p(z). Dewan et al proved that if p(z) has all its zeros in | z|≤ k, (k≤ 1), with s-fold zeros at the origin then for every α ∈ ℂ with | α |≥ kand for each r>0,In this paper, we obtain a refinement for above inequality. Also we extend some in-equalities for a polynomial of the form .

AbstractWe use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of type𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑. Rochon [Fractals 8: 355–368, 2000] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form 𝑤2+ 𝑐 is connected. We prove that our generalized Mandelbrot set of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑, is connected.