Brannan conjecture and sharp bounds on trigonometric sums

We discuss the Brannan Conjecture, which is an inequality on the coefficients of odd index of a certain power series. The conjecture was introduced in the current format in 1973 and had only been verified for odd indices 3, 5 and 7. In his master’s thesis the author was able to verify the Brannan Conjecture for all odd numbers up to 51. Here we first generalize the problem by introducing a new radius variable and give an integral representation of the problem which allows us to reduce the conjecture to a conjecture on a less complex coefficient. Then we prove the real version of our own conjecture which implies the real version of the Brannan conjecture. Also we prove our own conjecture for half of the domain for its radius variable which implies the Brannan conjecture for similar conditions.