For continuous vector-valued functions, we discuss relations among exponential and polynomial growth orders of the $\gamma$-Cesàro mean ($\gamma \ge 0$) and of the Abel mean. In general, the Abel mean has growth order not larger than those of Cesàro means, and a higher-order Cesàro mean has a smaller growth order than a lower-order Cesàro mean. But, for a positive function in a Banach lattice, the Abel mean and all $\gamma$-Cesàro means with $\gamma \ge 1$ (but not with $0 \le \gamma \lt 1$) have the same polynomial growth order. The possibility of non-equal growth orders for these means is illustrated by some examples of $C_0$-semigroups and cosine operator functions.
"GROWTH ORDERS OF CESÀRO AND ABEL MEANS OF FUNCTIONS IN BANACH SPACES." Taiwanese J. Math. 14 (3B) 1201 - 1248, 2010. https://doi.org/10.11650/twjm/1500405913