Consider a unit norm $\|V\|_2=1$ and a symmetric matrix $A$.

I wish to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$.

My belief is that this is true is motivated by empirical findings, using the following script (MATLAB):

 t=0.1;
 for ind=1:100000
    A=randn(5);A=A+A';
    v=randn(5,1); v=v/norm(v);    
    if norm(A^t*v)>   norm(A*v)^t
      sprintf('Claim does not hold')
      break
    end
 end
 sprintf('Claim holds')

Thanks a lot!