An evolutionary approach to the coefficient problems in the class of starlike functions

THEOREM 2.1. For every f ∈A, the following equivalence holds: f ∈S∗if and only if (2.1) Rezf ′(z) f(z) > 0, z ∈D \ {0}. Denote by P the subclass of Hol(D) of all analytic functions p having a positive real part on D given by (2.2) p(z) = 1 + ∞ X n=1 cnzn, z ∈D. It is well known ([6], cf. [16, Vol. I, p. 80]) that for every p ∈P of the form (2.2), the

THEOREM 2.3. Let m ∈N and Φ : Cm(P) →C be a continuous function, analytic in the interior of Cm(P). If a function p ∈P maximizes Re Φ on Cm(P), then p is of the form (2.4), i.e., p ∈Pm. From (2.4) and (2.6), the next lemma follows:

LEMMA 2.4. Let m ∈N. For p ∈Pm of the form (2.2), it holds that (2.7) cn = 2 m X j=1 λjxn j , n ∈N, where the xj are distinct points of T and λj ≥0, for j = 1, . . . , m, satisfy (2.5). 3. Differential evolution. Evolutionary algorithms are among the best general meth- ods for optimization. Differential evolution is based on population evolution. It has four components: • initialization, • mutation,

THEOREM 5.1. If f ∈S∗, then (5.1) | det H4,1(f)| ≤1 8 = 0.125. The inequality (5.1) is sharp and equality holds for the function f ∈S∗defined by zf ′(z) f(z) = 1 + z4 1 −z4 , z ∈D, and its rotations, i.e., for f(z) = z √ 1 −z4 , z ∈D,

THEOREM 5.2. If f ∈S∗is of the form (1.1), then |an| ≤n, n ∈Z2; (5.2) |Jm,n(f)| = |am+n−1 −anam| ≤(m −1)(n −1), m, n ∈Z2; (5.3) |J3,3,2(f)| = |a2a4 −a2 3| ≤1; (5.4) | det H3,1(f)| ≤4 9. (5.5) All inequalities are sharp. REMARK 5.3. The sharp estimates (5.2) were found by Nevanlinna [33] (cf. [13, p. 44]).