In the paper E. R. Lorch: A Curvature Study of Convex Bodies in Banach Spaces from 1953, the following assumptions and definitions are stated in section II (p. 107-108):
Let $(B, \| \cdot \|)$ be a real Banach space and $B^*$ its dual space. Let $r > 1$ and $G(x) := \frac{1}{r} \| x \|^r$ and $\phi(\alpha, \beta) := G(x + \alpha y + \beta z)$, where $x, y, z \in B$ are fixed and $\alpha, \beta \in \mathbb{R}$. Assuming that $\phi$ is twice continuously differentiable, the derivative of $\phi$ with respect to $\alpha$ at $\alpha = \beta = 0$ will be denoted the $G_y(x)$. [...] It is clear that $G_x(x) = r G(x)$. [...] The second derivatives of $\phi$ at $\alpha = \beta = 0$ will be denoted by $G_{y, y}(x)$, $G_{y, z}(x)$ and $G_{z, z}(x)$. [...] It is clear that $G_{y, z}(x) = G_{z, y}(x)$.
I am trying to understand the functions $G_y$, $G_{y, y}$, $G_{y, z}$ and $G_{z, z}$. Firstly, we have $G \colon B \to \mathbb{R}$ and $\phi \colon \mathbb{R} \times \mathbb{R} \to \mathbb{R}$.
I would write $$ G_y(x) = \frac{\partial}{\partial \alpha}\bigg|_{\alpha = \beta = 0} \phi(\alpha, \beta) = \lim_{\gamma \to 0} \frac{\phi(\gamma, 0) - \phi(0, 0)}{\gamma} = \lim_{\gamma \to 0} \frac{G(x + \gamma y) - G(x)}{\gamma} = \text{d}G(x; y), $$ where the last expression is the Gateaux derivative of $G$ at $x$ in direction $y$. This would make sense since we can then verify $G_x(x) = r G(x)$ (which is stated as "clear" above): for $x \in B$ we have $$ G_x(x) = \lim_{\gamma \to 0} \frac{G(x + \gamma x) - G(x)}{\gamma} = G(x) \lim_{\gamma \to 0} \frac{(1 + \gamma)^r - 1}{\gamma} = r G(x). $$ Furthermore, $y \mapsto G_y(x)$ is a linear bounded map $B \to \mathbb R$, which seems to be what is stated in theorem 1 ("If $x$ is any fixed element in $B$, then $G_y(x)$ represents a bounded linear functional").
How do the corresponding expressions for $G_{y, y}(x)$, $G_{y, z}(x)$ and $G_{z, z}(x)$ look like? In particular, are the second derivatives both taken with respect to $\alpha$ or are they mixed?
Wikipedia gives the following as one definition of the second order Gateaux derivative of $G$: $$ D^2 G(x)\{y ,z \} := \lim_{\gamma \to 0} \frac{\text{d}G(x + \gamma y; z) - \text{d}G(x; z)}{\gamma} = \frac{\partial^2}{\partial \alpha \partial \beta} \bigg|_{\alpha = \beta= 0} \phi(\alpha, \beta). $$ Is this expression $G_{y, z}(x)$?
If so, we would presumably have $$ G_{y, y}(x) = \frac{\partial^2}{\partial \alpha^2} \phi(\alpha, \beta) \bigg|_{\alpha = \beta = 0} = \frac{\partial^2}{\partial \alpha^2} G(x + \alpha y + \beta z) \bigg|_{\alpha = \beta = 0} = \lim_{\gamma \to 0} \frac{\text{d}G(x + \gamma y; y) - \text{d}G(x; y)}{\gamma} $$ and analogously $$ G_{z, z}(x) = \frac{\partial^2}{\partial \beta^2} \phi(\alpha, \beta) \bigg|_{\alpha = \beta = 0} = \frac{\partial^2}{\partial \beta^2} G(x + \alpha y + \beta z) \bigg|_{\alpha = \beta = 0}. $$
According to Theorem 2 in that paper, we have $G_{y, z}(x) \in B^*$ for $y \in B$ and $G_y(x) \in B^*$ for $x \in B$. Do my above guesses about $G_{y, y}$, ... fulfill those requirements?