Equation goes outside the boundary with eqnarray environment!

You have to break the too-long equation manually. Here's a way to do so it seems natural, using some spacing adjustment via \phantom:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{align*}
  \alpha + M(d(fx_n,fz_n)) &\leq \alpha + \alpha(fx_n,fz_m) M(d(fx_n,fz_k)) \\
                           &\leq M(M_{f,h}(fx_n,fz)) \\
                           &= M(x,y) \max \bigl\{ d(fx_n,fz),d(fx_n,fx_n), \\
                           &\phantom{{}={} M(x,y) \max \bigl\{}d(fz,fz),\tfrac{d(fx_n,fz)+d(fz_b,fx_n)}{h} \bigr\}
\end{align*}

\end{document}

Please avoiding using eqnarray.

Another align*-based solution. It differs from @Werner's solution in the way the final row's indentation is achieved and in the treatment of the fractional expression in the final row.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
\alpha+M(d(fx_n,fz_n))
&\leq\alpha+\alpha(fx_n,fz_m)M(d(fx_n,fz_k))\\
&\leq M(M_{f,h}(fx_n,fz))\\
&= M(x,y)\max\bigl\{ d(fx_n,fz),d(fx_n,fx_n),\\
&\qquad\qquad d(fz,fz),\tfrac{1}{h}\bigl[d(fx_n,fz)+d(fz_b,fx_n)\bigr] \bigr\}
\end{align*}
\end{document}

I'll add three other variants:

\documentclass{article}
\usepackage[showframe]{geometry}%
 \usepackage{mathtools, nccmath}

\begin{document}

\begin{align*}
  \alpha+M(d(fx_n,fz_n))
    & \leq\alpha+α(fx_n,fz_m)M(d(fx_n,fz_k)) \\
    & \leq M(M_{f,h}(fx_n,fz)) \\
    & =\!\begin{multlined}[t] M(x,y)\max\bigl\{ d(fx_n,fz),d(fx_n,fx_n), d(fz,fz), \\[0.5ex]
  \mfrac{1}{h}\bigl[d(fx_n,fz)+d(fz_b,fx_n)\bigr] \bigr\}
  \end{multlined}
\end{align*}

\begin{align*}
  \alpha+M(d(fx_n,fz_n))
  &\leq\alpha+α(fx_n,fz_m)M(d(fx_n,fz_k))\\
  &\leq M(M_{f,h}(fx_n,fz))\\
    & =\!\begin{aligned}[t] M(x,y)\max\bigl\{ d(fx_n,fz),d(fx_n,fx_n), d(fz,fz) & , \\
  \mfrac{1}{h}\bigl[d(fx_n,fz)+d(fz_b,fx_n) & \bigr] \bigr\}
  \end{aligned}
\end{align*}

\begin{align*}
  \alpha+M(d(fx_n,fz_n))
    & \leq\alpha+α(fx_n,fz_m)M(d(fx_n,fz_k)) \\
    & \leq M(M_{f,h}(fx_n,fz)) \\
    & =\! M(x,y)\max \!\begin{Bmatrix*}[r]d(fx_n,fz),d(fx_n,fx_n), d(fz,fz), \\[0.5ex]
  \mfrac{1}{h}\bigl[d(fx_n,fz)+d(fz_b,fx_n)\bigr]
  \end{Bmatrix*}
\end{align*}

\end{document}