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

In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this purpose, we make use of a combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we derive the estimates on the initial Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in these analytic and bi-concave function classes, which are introduced in this paper. We also derive a number of corollaries and consequences of our main results in this paper.

Institute of Mathematics of the Czech Academy of Sciences

In this paper we examine the closure properties of the class $\mathcal{V}_\mathcal{H}(F;\gamma)$ under the generalized Bernardi-Libera-Livingston integral \\operator $\mathcal{L}_{c}(f),$ ($c>-1$) which is defined by $ \mathcal{L}_{c}(f)=\mathcal{L}_{c}(h)+\overline{\mathcal{L}_{c}(g)}$ where \begin{equation*} \mathcal{L}_{c}(h)(z)=\frac{c+1}{z^{c}}\int\limits_{0}^{z}(t^{c-1}h(t) dt \;\;\;% \mathrm{and} \;\;\; \mathcal{L}_{c}(g)(z)=\frac{c+1}{z^{c}}\int\limits_{0}^{z}(t^{c-1}g(t) dt. \end{equation*} The obtained results are sharp and they improve known results.

In this paper, we consider a class L(λ, μ; ϕ) of analytic functions f defined in the open unit disk U satisfying the subordination condition that , where is the Sălăgean operator and ϕ(z) is a convex function with positive real part in U. We obtain some characteristic properties giving the coefficient inequality, radius and subordination results, and an inclusion result for the above class when the function ϕ(z) is a bilinear mapping in the open unit disk. For these functions f (z) ; sharp bounds for the initial coefficient and for the Fekete-Szegö functional are determined, and also some integral representations are given.