Hardy-Littlewood Maximal Function and Satisfied Cricketer 

The maximal problem

In the words of Hardy and Littlewood, two renowned mathematicians and ardent followers of cricket in their original paper gave an analogy in terms of cricket.

The problem is most easily grasped when stated in the language of cricket or any other game in which a player compiles a series of scores of which an average is recorded. Suppose that a batsman plays, in a given season, a given ‘stock’ of innings 

say b1,b2,…, bn determined in everything except arrangement. Let ai be his maximum average for any consecutive series of innings ending in the ith, so that

we may agree that, in case of ambiguity, i* is chosen to be as small as possible.

Lets s(x) be a positive function which increases with x, and let his ‘satisfaction’ after the ith innings be measured by si = s(ai). Finally, let his total satisfaction for the season be measured by 

The total satisfaction S is maximum, for a given stock of innings, when the innings are played in decreasing order. The proof we leave it to the reader .

The result above is the discrete form of the celebrated maximal theorem of
Hardy and Littlewood;

Consider the timestamp of recently concluded India Australia Adelaide test.

If you want to read more about the match follow the below link.

Coming back to the question of satisfied cricketer, we will use this analogy to compare how satisfied were the top five Indian batsmen. Prithvi Shaw will serve as a limiting batsman as he has played the least number of inns, nine to be precise. The cut off of a number of inns is the last nine inns for remaining folks as start Mayank, Pujara, Kohli, and Rahane, which boils down to tests from Ranchi to Adelaide.

Prithvi Shaw’s last 9 inns score is

134, 70, 33, 16, 14, 54, 14, 0, 4

(assuming 33 as completed inns for simplicity). Other four didn’t have any unbeaten knock in this phase of career.

We define a index called as satisfaction which is average after each knock.

Shaw’s satisfaction list will be

134.0, 102.0, 79.0, 63.25, 53.4, 53.5, 47.86, 41.88, 37.67

If you sum this satisfaction you will get overall satisfaction, Shaw’s overall satisfaction is 612.55

The table for overall satisfaction of top five batsman.

Overall Satisfaction Score

This number is a reflection of the fact decreasing scores have higher satisfaction than increasing scores. Hardy’s Maximal function tells batsman should be more disturbed if he keeps on improving. Prithvi Shaw has taken this statement literally in his armor and thought process.