For f(R)=R+αR^n, the dual scalar-tensor potential is bounded below with a minimum exactly for even n and positive α, matching the f''(R)>0 condition.
f (r) theories of gravity.Reviews of Modern Physics, 82(1):451–497, 2010
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Does the stability of f(R) theories imply the stability of the dual scalar-tensor theory ?
For f(R)=R+αR^n, the dual scalar-tensor potential is bounded below with a minimum exactly for even n and positive α, matching the f''(R)>0 condition.