A Special Class of Univalent Functions in Hele-Shaw Flow Problems

and Applied Analysis 3 The equation satisfied by the free boundary is 4, 6 Re [ Ḟ ζ, t ζF ′ ζ, t ] Q 2π , ζ e 1.5 for the zero tension surface model and Re [ Ḟ ζ, t ζF ′ ζ, t ] Q 2π − γH [ i ∂k ∂t ( e, t )] θ , ζ e 1.6 for the small surface tension model. 2. The Inner Problem (Bounded Domains) In this section, we obtain the invariance in time ofΦ-likeness property for the inner problem. Starting with an initial bounded domainΩ 0 which is Φ-like, we prove that at each moment t ∈ 0, T the domain Ω t is Φ-like both for zero and nonzero surface tension models . Definition 2.1. Let f be a holomorphic function onU such that f 0 0 and f ′ 0 / 0. LetΦ be a holomorphic function on f U such that Φ 0 0 and Re Φ′ 0 > 0. We say that f is Φ-like on U or Φ-like if Re [ zf ′ z Φ ( f z ) ] > 0, for each z ∈ U. 2.1 We remark that a Φ-like function is univalent. In fact, any univalent function is Φ-like for some Φ. Remark 2.2. a The concept of Φ-likeness was introduced and studied by Brickman in 1973 12 and generalizes the notions of starlikeness and spiral-likeness. Applications of this notion in the study of univalence may be found in 13 . b If Φ w w in the above definition, then f is starlike. c If Φ w λw and Re λ > 0, then f is spiral-like of type − argλ. We restate that a holomorphic function f on U such that f 0 0 and f ′ 0 / 0 is said to be spiral-like of type α ∈ −π/2, π/2 if Re ezf ′ z /f z > 0, z ∈ U 13, 14 . The following result is a generalization of 1, Theorem 1 to the case of Φ-like functions. The mentioned theorem may be obtained by taking Φ w ≡ w in Theorem 2.3 below. Theorem 2.3. Let Q < 0 and f0 be a function which is Φ-like on U and univalent on U. Let f ζ, t be the classical solution of the Polubarinova-Galin equation 1.1 with the initial condition f ζ, 0 f0 ζ . Also let Ω ⋃ 0≤t 0, ∀w ∈ Ω, 2.2 then f ζ, t is Φ like for t ∈ 0, T . 4 Abstract and Applied Analysis Proof. Taking into account that all the functions f ζ, t have analytic univalent extensions to U for each t ∈ 0, T and in consequence their derivatives f ′ ζ, t are continuous and do not vanish inU, we can replace with “≥” the inequality in the definition 2.1 of aΦ-like function. The equality can be attained only for |ζ| 1. We suppose by contrary that the conclusion of Theorem 2.3 is not true. Then there exist t0 ≥ 0 and ζ0 e0 such that arg ζ0f ′ ζ0, t0 /Φ f ζ0, t0 π/2 or − π/2 , which is equivalent to Re ζ0f ′ ζ0, t0 Φ ( f ζ0, t0 ) 0, Im ζ0f ′ ζ0, t0 Φ ( f ζ0, t0 ) / 0, 2.3 and for each ε > 0, there are t > t0 and θ ∈ θ0 − ε, θ0 ε such that