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In this paper we show that, for any fixed $1<c<\frac{5363}{3900}$, everysufficiently large positive number $N$ and a small constant $\varepsilon>0$,the diophantine inequality \begin{equation*}|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c-N|<\varepsilon \end{equation*} has a solution inprime numbers $p_1,\,p_2,\,p_3,\,p_4,\,p_5$, such that $p_1=x^2 + y^2 +1$.

A quinary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$