Let f : V → W be C2, where V and W are normed vector spaces. Define

M_0 = sup{∥f(x)∥_w : x ∈ V}

M_1 = sup{∥f’(x)∥_L(v,w) : x∈V}

M_2 = sup{∥f”(x)∥_L2(v,w) : x∈V}

Assume that these are all finite. Show that

M_1^2 ≤ 4M_0M_2.

[Hint: use Taylor’s theorem to show that for any x∈V and v∈V, and any s>0,

∥f’(x)(v)∥W ≤ (∥f(x+sv) – f(x)∥W)/s + (s/2)M_2,

And so deduce ∥f’(x)∥L(v,w) ≤ 2M_0/s + sM_2/2. Choose s carefully.]

could anyone please help me with this problem ??