The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
2. Explicitly ORDER BY After GROUP BY (modified query below): By default, the database sorts all 'GROUP BY col1, col2, ...' queries as if you specified 'ORDER BY col1, col2, ...' in the query as well. If a query includes a GROUP BY clause but you want to avoid the overhead of sorting the result, you can suppress sorting by specifying 'ORDER BY NULL'.
3. Index Function Calls Using Generated Columns (modified query below): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index to optimize the search. Creating and indexing a generated column (supported in MySQL 5.7) will allow MySQL to optimize the search.
4. Remove Redundant Left Joins (modified query below): Redundant LEFT JOINs (e.g. `invoice`) were detected in the query. Removing them will result in a performance improvement. In some cases, JOINs become redundant after an optimization is applied, such as when converting OR conditions to a UNION clause.

Optimal indexes for this query:

ALTER TABLE `invoice` ADD INDEX `invoice_idx_postdate` (`postdate`);
ALTER TABLE `invoice` ADD INDEX `invoice_idx_date_postdate` (`date_postdate`);

The optimized query:

SELECT
        SQL_CACHE date(todaydata.postdate) AS postdate,
        SUM(CASE 
            WHEN todaydata.amount > 0 THEN todaydata.amount 
            ELSE 0 END) AS INFLOW,
SUM(CASE 
    WHEN todaydata.amount < 0 THEN todaydata.amount 
    ELSE 0 END) AS OUTFLOW 
FROM
invoice AS todaydata 
WHERE
todaydata.postdate BETWEEN now() - INTERVAL 2 month AND now() + INTERVAL 1 month 
GROUP BY
todaydata.date_postdate 
ORDER BY
NULL