From Olver's "Applied Linear Algebra":
Suppose $\langle v,w \rangle_1$ and $\langle v,w \rangle_2$ are two inner products on the same vector space V. For which $\alpha, \beta \in \mathbb{R} $ is the linear combination $\langle v,w \rangle = \alpha \langle v,w \rangle_1 + \beta \langle v,w \rangle_2$ a legitimate inner product? Hint: The case $\alpha, \beta \geq 0$ is easy. However, some negative values are also permitted, and your task is to decide which.
Bilinearity and Symmetry are automatically guaranteed for any $\alpha, \beta$ so we just need positivity:
$\alpha\langle v,v\rangle_1+ \beta\langle v,v\rangle_2 > 0$ for all $v \neq 0 \in V$
Obviously this is true if $\alpha, \beta$ are both $\geq 0$. If $\beta = 0$ then it's true when $\alpha > 0$ (likewise for the mirror case). Third possibility is $\alpha > 0, \beta < 0$.
We can rearrange the equation to $\langle v,v \rangle_1 > -\frac{\beta}{\alpha} \langle v,v \rangle_2$ but that doesn't help
How to proceed?
Define the unit sphere $C_{1}$ with respect the first inner product,i.e $C_{1}=\{v \in V \backslash \langle v,v\rangle_{1}=1\}$,assume $\beta <0, \alpha >0 $ Denote by $M:=\max \{\langle v,v\rangle_{2} \backslash v\in C^{1}\}$,then $\forall v\in C_{1}:\alpha +\beta \langle v,v\rangle_{2} \geq \alpha+\beta M$,So a sufficient condition for positivity is $\alpha+\beta M >0 \leftrightarrow \beta >\frac{-\alpha}{M}$ ,in the same way define $C_{2}=\{v \in V \backslash \langle v,v\rangle_{}=1\}$;Denote by $m:=\min \{\langle v,v\rangle_{1} \backslash v\in C_{2}\}$,then u can verify that $\beta > \frac{-\alpha}{m}$ is another sufficient condition, now let $L=\min\{-\alpha m,\frac{-\alpha}{M} \}$,the the combination defines an inner product iff $\beta > L$ (the necessity follows from the compactness of $C_{1}$ and $C_{2}$)