Abstract: In the paper Lu (2011) [1], the maximum principle was used to study the uniformly bounded $L^{\infty}$ estimates $z(\rho^{\delta,\varepsilon},u^{\delta,\varepsilon}) \leq B(x)$, $w(\rho^{\delta,\varepsilon},u^{\delta,\varepsilon}) \leq M(t)$ for the $\varepsilon$-viscosity and $\delta$-flux-approximation solutions $(\rho^{\delta,\varepsilon},u^{\delta,\varepsilon})$ of the nonhomogeneous system (\ref{1.3}), where $w$, $z$ are Riemann invariants of (\ref{1.3}) and $ M(t) $ depends on the bound of the nonlinear function $a(x)$, which excludes the class of discontinuous functions. In this short paper, we obtain the estimate $w(\rho^{\delta,\varepsilon},u^{\delta,\varepsilon}) \leq \beta$ when $a'(x) \geq 0$ for a suitable constant $\beta$ depending only on the bound of $a(x)$ and prove the existence of bounded entropy solutions, for the Cauchy problem of the isentropic Euler equations with geometric effects (\ref{1.1}), which extend the results of finite energy solution in [2], and weak solutions in [3] for a polytropic gas with $\gamma \in (1, \frac{5}{3}]$ to the general pressure function $P(\rho)$. $$\left\{ \label{1.1} \begin{array}{l} ( \rho a(x))_{t}+(\rho u a(x))_{x}=0, \\ ( \rho u a(x))_t+( \rho u^2 a(x))_x+ a(x) P(\rho)_x=0 \end{array}\right.\tag{1.1} $$ $$\left\{ \label{1.3} \begin{array}{l} \rho_{t}+(\rho u)_{x}= - \frac{a'(x)}{a(x)} \rho u \\ ( \rho u)_t+( \rho u^2+ P(\rho))_x= - \frac{a'(x)}{a(x)} \rho u^{2}. \end{array}\right.\tag{1.3} $$