The idiom on the up and up—whose exact origins are mysterious, though it dates from the late 19th century, is likely American, and appears to come from sports betting—means (1) open and honest, legitimate; and (2) on the rise. Though some dictionaries emphasize one definition or the other (usually the first), the idiom is widely used in both senses.
Some writers hyphenate the last three words of the phrase—on the up-and-up—but there’s no good reason to do this.
In most pre-1950 examples of on the up and up, the phrase means open and honest—for example:
Expert boxing writers estimate 90 per cent of all fights are on the up and up. [Evening Independent (1938)]
Other presidents have quizzed Judges before appointment to make sure they were on the up-and-up, notably Theodore Roosevelt and Woodrow Wilson. [The Day (1937)]
A guy might be known as a square shooter, on the up and up, or he might be a rat. [Oxnard Daily Courier (1929)]
But there are a few pre-1950 examples of on the up and up used to mean on the rise—for example:
Country centres where industries have started are pleading for them, but with costs on the up and up and money hard to raise, extensions are difficult. [Sydney Morning Herald (1949)]
But these examples are rare (and there are none easily found in U.S. sources), so the on the rise definition of on the up and up is probably a newer arrival.
In 21st-century writing, the phrase is very often used to mean on the rise—for example:
Twitter’s value is on the up-and-up … and up. [New York Magazine]
Some areas of town still a no-go zone but things are on the up-and-up. [New Zealand Herald]
This was a university on the up-and-up, I was told. Come and be part of it. [Las Vegas Sun]
Yet the original sense is still used—for example:
All that aside, Redd and others say they simply want to make sure the whole process is on the up and up. [Madison County Journal]
Not to worry; certainly developers and builders can be trusted to do things completely on the up and up without much in the way of oversight. [The Gazette]
Perry said he did independent auditing of the company in its heyday and that everything was on the up and up. [Houston Chronicle]