Word or phrase for non-linear-but-still-greater-than-linear?

Solution 1:

The term for what you're asking is superlinear, although it may not be understood depending on your audience.

From Wiktionary:

Adjective superlinear ‎(comparative more superlinear, superlative most superlinear)

1. Above a line
2. (mathematics) Describing a function that eventually grows faster than any linear one

Merriam Webster redirects superlinear to supralinear. Unfortunately, the full definition of this word is only available in the unabridged version of MW, which is not free.

But, I did find a definition of supralinear at Wiktionary:

Adjective supralinear ‎(not comparable)

1. (typography) above the lines of normal text
2. (mathematics) greater than linear

You might be better off phrasing things a different way, like saying that the growth rate is 'accelerating', or using "more and more" or something similar.

Solution 2:

Perhaps you could say that "development of new technology in this field is accelerating".

In my view, "accelerating development" would mean that there was more development in 2015 that there was in 2010.

Solution 3:

You are probably looking for polynomial. Linear functions are polynomial, but are the slowest growing polynomials. All polynomials (eventually) grow more slowly than exponentials.