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In this paper, we investigate the $\left[p,q\right]-\phi$ order and $\left[p,q\right]-\phi$ type of $f_1+f_2$, $f_1f_2$ and $\frac{f_1}{f_2}$, where $f_1$ and $f_2$ are analytic or meromorphic functions with the same $\left[p,q\right]-\phi$ order and different $\left[p,q\right]-\phi$ type in the unit disc. Also, we study the $\left[p,q\right]-\phi$ order and $\left[p,q\right]-\phi$ type of different $f$ and its derivative. At the end, we investigate the relationship between two different $\left[p,q\right]-\phi$ convergence exponents of $f$. We extend some earlier precedent well known results.
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