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We give an equality condition for a symmetrization inequality for condensers proved by F.W. Gehring regarding elliptic areas. We then use this to obtain a monotonicity result involving the elliptic area of the image of a holomorphic function f.

We give an explicit relation between the slope of the trajectory of a semigroup of holomorphic functions and the harmonic measure of the associated planar domain Ω. We use this to construct a semigroup whose slope is an arbitrary interval in [−π/2, π/2]. The same method is used for the slope of a backward trajectory approaching a super-repulsive fixed point.

Let X ⊂ ℝ be a bounded set; we introduce a formula that calculates the upper graph box dimension of X (i.e. the supremum of the upper box dimension of the graph over all uniformly continuous functions defined on X). We demonstrate the strength of the formula by proving various corollaries. We conclude by constructing a collection of sets X with infinitely many isolated points, having upper box dimension a taking values from zero to one while their graph box dimension takes any value in [ max {2a, 1}, a + 1], answering this way, negatively to a conjecture posed.1

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