Let \(A\) denote the class of all normalized analytic functions \(f\) \((f(0)=0=f^{\prime}(0)-1)\) in the open unit disc \(\Delta\). For \(0<\lambda\leq 1\), define \[ U(\lambda )=\{ f\in A\;:\;\left| \left( \frac{z}{f(z)}\right) ^2 f^{\prime}(z)-1\right| <\lambda\;,\;z\in \Delta \} \] and \[ P(2\lambda )=\{ f\in A\;:\;\left| \left( \frac{z}{f(z)}\right) ^{\prime\prime} \right| <2\lambda\;,\;z\in \Delta \}\;. \] Recently, the problem of finding the starlikeness of these classes has been considered by Obradovic and Ponnusamy, and later by Obradovic {et al}. In this paper, the authors consider the problem of finding the order of starlikeness and convexity of \(\;U(\lambda )\) and \(P(2\lambda )\), respectively. Theorem 1.2 If \(f\in U(\lambda )\) and \(a=| f^{\prime\prime}(0)| /2 \leq 1\), then \(f\in S^{*}(\delta )\) whenever \(0<\lambda \leq \lambda (\delta )\), where \[ \lambda (\delta )={\frac{\sqrt{(1-2\delta )(2-a^2-2\delta )}-a(1-2\delta )}{2(1-\delta )}}\;\;\;\text{ if }\;0\leq \delta <\frac{1+a}{3+a} \] and \[ \lambda (\delta )={\frac{1-\delta (1+a)}{1+\delta }} \;\;\;\text{ if }\;\frac{1+a}{3+a}\leq\delta <\frac{1}{1+a}\;. \] Theorem 1.5 Let \(f\in A\) with \(f^{\prime\prime}(0)=0\) and suppose that \[ {\left| \left( \frac{z}{f(z)}\right)^2 f^{\prime}(z)\left( 1+\frac{zf^{\prime\prime}(z)}{2f^{\prime}(z)} -\frac{zf^{\prime}(z)}{f(z)}\right) \right| <\lambda\;,\;z\in\Delta} \] or equivalently \[ { \left| z^2 \left( \frac{z}{f(z)}\right) ^{\prime\prime} \right| <2\lambda\;,\;z\in\Delta}\;, \] for some \(0<\lambda\leq \frac{1}{\sqrt{2}}\). Then \(f\in K(\beta )\), where \[ \beta =\beta (\lambda )= { \frac{1+\lambda ^2-6\lambda}{1-\lambda ^2}}\;\;\;\text{ for }\;0<\lambda \leq\frac{1}{2} \] and \[ \beta =\beta (\lambda )= { \frac{-\lambda (2+3\lambda )}{1-\lambda ^2}\;\;\;} \text{ for }\;\frac{1}{2}<\lambda \leq\frac{1}{\sqrt{2}}\;. \] In particular, \(P(2\lambda )\subset K\) if \(0<\lambda\leq 3-2\sqrt{2}\;.\) In addition to these results, the authors provide a coefficient condition for a function to be in \(K(\beta )\) and propose a conjecture that each function \(f\in U(\lambda )\) with \(f^{\prime\prime} (0)=0\) is convex at least when \(0<\lambda\leq 3-2\sqrt{2}\).