The real and complex convexity

We prove that the holomorphic differential equation $\varphi^{\prime \prime}(\varphi+c) = \gamma(\varphi^{\prime})^{2} (\varphi:\mathbb{C}\rightarrow \mathbb{C}$ be a holomorphic function and $(\gamma, c) \in \mathbb{C}^{2})$ plays a classical role on many problems of real and complex convexity. The condition exactly $\gamma \in \{1, \frac{s-1}{s} \/ s \in \mathbb{N} \backslash \{0\}\}$ (independently of the constant c) is of great importance in this paper. On the other hand, let $n \geq 1, (A_{1}, A_{2}) \in \mathbb{C}^{2}$ and $g_{1}, g_{2} : \mathbb{C}^{n} \rightarrow \mathbb{C}$ be two analytic functions. Put $u(z, w) = \| A_{1}w - g_{1}(z) \|^{2} + \| A_{2}w - g_{2}(z) \| ^{2}v(z,w) = \| A_{1}w - \overline{g_{1}}(z) \| ^{2} + \| A_{2}w - \overline{g_{2}}(z) \|^{2}$, for $(z,w) \in \mathbb{C}^{n} \times \mathbb{C}$. We prove that $u$ is strictly plurisubharmonic and convex on $\mathbb{C}^{n} \times \mathbb{C}$ if and only if $n = 1, (A_{1}, A_{2}) \in \mathbb{C}^{2} \backslash \{0\}$ and the functions $g_{1}$ and $g_{2}$ have a classical representation form described in the present paper. Now $v$ is convex and strictly psh on $\mathbb{C}^{n} \times \mathbb{C}$ if and only if $(A_{1}, A_{2}) \in \mathbb{C}^{2} \backslash \{0\}, n \in \{1,2\}$ and and $g_{1}, g_{2}$ have several representations investigated in this paper