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Jan 17, 2018 at 22:15 comment added Iosif Pinelis It is possible, by strengthening the argument a bit. I have posted an answer to your additional question at mathoverflow.net/questions/290965/… .
Jan 17, 2018 at 4:10 comment added H_R Do you know if it's possible to control (in terms of $a$) the smallest $x$ such that $f(x) \leq 0$?
Jan 12, 2018 at 14:51 comment added Iosif Pinelis I have updated the answer to show more clearly that actually more is proved than requested.
Jan 12, 2018 at 14:48 history edited Iosif Pinelis CC BY-SA 3.0
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Jan 12, 2018 at 5:37 vote accept H_R
Jan 12, 2018 at 5:36 comment added H_R Oh I'm sorry, I didn't see that $R$ could be infinite.
Jan 12, 2018 at 4:39 comment added Iosif Pinelis @H_R : Because $r$ is positive and concave and hence nondecreasing (make a picture). I have also added a detail on this.
Jan 12, 2018 at 4:37 history edited Iosif Pinelis CC BY-SA 3.0
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Jan 12, 2018 at 1:32 comment added H_R Why does $r(x) \to R$ as $x \to \infty$?
Jan 11, 2018 at 20:37 history edited Iosif Pinelis CC BY-SA 3.0
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Jan 11, 2018 at 20:32 history edited Iosif Pinelis CC BY-SA 3.0
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Jan 11, 2018 at 20:06 comment added Iosif Pinelis It looks like I have been able to improve the previous partial affirmative answer to a complete one now
Jan 11, 2018 at 20:06 history edited Iosif Pinelis CC BY-SA 3.0
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Jan 11, 2018 at 6:28 history edited Iosif Pinelis CC BY-SA 3.0
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Jan 11, 2018 at 3:57 history edited Iosif Pinelis CC BY-SA 3.0
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Jan 11, 2018 at 2:27 history answered Iosif Pinelis CC BY-SA 3.0