By using the q-derivative operator and Legendre polynomials, some new subclasses of q-starlike functions bi-univalent are introduced. Several coefficient estimates Fekete–Szegö-type inequalities for in each these obtained. The results derived this article shown to extend generalize those earlier works.

In current manuscript, using Laguerre polynomials and (p−q)-Wanas operator, we identify upper bounds a2 a3 which are first two Taylor-Maclaurin coefficients for a specific bi-univalent functions classes W∑(η,δ,λ,σ,θ,α,β,p,q;h) K∑(ξ,ρ,σ,θ,α,β,p,q;h) cover the convex starlike functions. Also, discuss Fekete-Szegö type inequality defined class.

In the current work, we use (M,N)-Lucas Polynomials to introduce a new family of holomorphic and bi-univalent functions which involve linear combination between Bazilevic ?-pseudo-starlike function defined in unit disk D establish upper bounds for second third coefficients belongs this family. Also, discuss Fekete-Szeg? problem

let $p$ be an analytic function defined on the open unit disc $mathbb{d}$ with $p(0)=1.$ the conditions on $alpha$ and $beta$ are derived for $p(z)$ to be subordinate to $1+4z/3+2z^{2}/3=:varphi_{c}(z)$ when $(1-alpha)p(z)+alpha p^{2}(z)+beta zp'(z)/p(z)$ is subordinate to $e^{z}$. similar problems were investigated for $p(z)$ to lie in a region bounded by lemniscate of bernoulli $|w^{2}-1...

In this paper we consider the classes of starlike functions, starlike functions of order α, convex functions, convex functions of order α and the classes of the univalent functions denoted by SH (β), SP and SP (α, β). On these classes we study the convexity and αorder convexity for a general integral operator.