In the present work, we define two new families of analytic and m-fold symmetric biunivalent functions in open unit disk $\Delta$. Also, for each classes introduced here, prove upper bounds initial coefficients $\left|b_{m+1}\right|$ $\left|b_{2 m+1}\right|$. Furthermore, get special cases our results.

This paper presents a new general subfamily NΣmu,v(η,μ,γ,ℓ) of the family Σm that contains holomorphic normalized m-fold symmetric bi-univalent functions in open unit disk D associated with Ruscheweyh derivative operator. For belonging to introduced here, we find estimates Taylor–Maclaurin coefficients am+1 and a2m+1, consequences results are discussed. The current findings both extend enhance ...

The theory of harmonic univalent mappings has become a very popular research topic in recent years. The aim of this expository article is to present a guided tour of the planar harmonic univalent and related mappings with emphasis on recent results and open problems and, in particular, to look at the harmonic analogues of the theory of analytic univalent functions in the unit disc.