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!