Summary: In this paper, among other things, we prove the following results.
Theorem 1: Let \(f(z)=z+a_ {n+1}z^ {n+1}+\cdots\) be analytic and satisfy \(| f^\prime(z)-1| <1-\alpha\), \(0\leq \alpha <1\), for \(| z| <1\). Then \(\operatorname{Re}\{e^ {i\beta} f(z)/z\}>0\), where \(| \beta| \leq \pi/2 -\arcsin ((1-\alpha)/(n+1))\). The result is sharp.
Theorem 4: Let \(p(z)\) be analytic and \(h(z)\) be convex univalent in \(| z| <1\) with \(p(0)=h(0)=1\). If \(f(z)\prec h(z)\), then \(p(I_ \alpha z)+\alpha r_ \alpha zp'(r_ \alpha z)\prec h(z)\), where \(r_ \alpha =\sqrt {1+\alpha^ 2}-\alpha\) and \(\alpha >0\).
Theorem 5: Let \(p(z)\) be analytic and \(h(z)=z+\cdots\) be convex univalent in \(| z| <1\). If \(cp(z)+zp^\prime(z)=\lambda h(z),c>0\), then (i) for every \(\lambda\), \(0<\lambda \leq \lambda_ c =\frac 12 (1/c-\int^ 1_ 0 (u^ {c-1}/(1+u))\,du)^ {-1}\), \(p(z)\prec h(z)\); (ii) for every \(\lambda\), \(\lambda >\lambda_ c, p(r_ 0z)\prec h(z)\), where \(r_ 0\) is the root in \((0,1)\) of the equation \(\int^ 1_ 0 (u^ {c-1}/(1+ru))\,du-1/c+1/(2\lambda) =0\). The results are sharp.