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Let $S$ denote the class of functions $f\left(z\right)=z+a_2{z}^2+a_3{z}^3+...$ univalent and holomorphic in the unit disc $𝛥=\left\{z\right|z|<1\}$. In the paper we obtain a sharp estimate of the functional $|a_3-\alpha a_2^2|+\alpha |a_2{|}^2$ in the class $S$ for an arbitrary $\alpha \in ℝ$.

Let $S$ denote the class of functions $f\left(x\right)=z+a_2{z}^2+a_3{z}^3+...$ univalent and holomorphic in the unit disc $\Delta =\{z:\left|z\right|<1\}$. In the paper we obtain an estimate of the functional $\left|a_3-c{a}_2^2\right|+c{\left|a_2\right|}^n$ in the class $S$ for arbitrarily fixed $x\in 𝐑$ and $n=1,2,3,...$. Hence, for some special values of the parameters, we obtain estimates of several interesting functionals and numerous applications. A few open problems of a similar type are also formulated.