Consider the function
$$f(x,y)=g_{1}(x,y)=-\frac{1}{y}\frac{\left( \ln \frac{x}{a}\right) ^{p}\left( \ln \frac{b}{y}\right) ^{p}}{\left( \ln \frac{b}{a}\right) ^{p}}, 1\leqslant a\leqslant x\leqslant y\leqslant b,\text{ }a\neq b$$ and $$f(x,y)=g_{2}(x,y)=g_{1}(x,y)+\frac{1}{y}\left( \ln \frac{x}{y}\right) ^{p}, 1\leqslant a\leqslant y\leqslant x\leqslant b,\text{ }a\neq b.$$ where $0<p\leqslant 1.$ Find $\underset{x,y\in \lbrack a,b]}{\sup }\left\vert f(x,y)\right\vert.$