On bounds for the derivative of analytic functions at the boundary

In this paper, we obtain a new boundary version of the Schwarz lemma for analytic function. We give sharp upper bounds for $\left\vert f^{\prime}(0)\right\vert $ and sharp lower bounds for $\left\vert f^{\prime}(c)\right\vert $ with $c\in \partial D=\left\{ z:\left\vert z\right\vert=1\right\} $. Thus we present some new inequalities for analytic functions. Also, we estimate the modulus of the angular derivative of the function $f(z) $ from below according to the second Taylor coefficients of $f$ about $z=0$ and $z=z_{0}\neq 0.$ Thanks to these inequalities, we see the relation between $\vert f^{\prime }(0)\vert$ and $\Re f(0).$ Similarly, we see the relation between $\Re f(0)$ and $\vert f^{\prime }(c)\vert$ for some $c\in\partial D.$ The sharpness of these inequalities is also proved.