open conjectures in real analysis targeting real valued functions of a single real variable

I am hoping that this question (if in acceptable form) be community wiki.

Are there any open conjectures in real analysis primarily targeting real valued functions of a single real variable ? (it may involve other concepts but primarily targeting these type of functions and no involvement of number theory). Its not that i am going to attack them right away ( not due to lack of interest but time) but am just curious to know them. Also how difficult it is get a new result in this area ?

Solution 1:

Khabibullin’s conjecture on integral inequalities. Also, according to this entry at the Open Problem Garden, the following question is open: Give a necessary and sufficient criterion for the sequence $a_n$ so that the power series $\sum a_nx^n$ is bounded for all $x$ in $\bf R$.

Solution 2:

Probably the quickest way to find a large number of problems of current interest in real analysis is to look through the journal "Real Analysis Exchange", especially the Queries section and the open problems listed in (most of) the survey articles.

(added October 28) Brian S. Thomson's book Symmetric Properties of Real Functions (Marcel Dekker, 1994) has several pages of problems at the end. Some of these problems have been open for a long time. Some of the other problems were simply questions that occurred to Thomson when he wrote the book and wasn't able to answer right off. Many of these problems have been answered by now, but I'm sure there remain many that are still unanswered. Since the topic of this book is rather specialized, the number and variety of problems serves well to indicate there is no lack of open problems targeting real valued functions of a real variable.