Greater than sudoku

Each of the numbers 1 to 9 must appear exactly once in each row, column, and block. In addition, adjacent cells must obey any "greater than" (>) or "less than" (<) symbol that appears on their dividing line. Note that this puzzle has no starting clues given; the greater than symbols are enough to determine a unique solution without any initial conditions at all! As an aid in solving, some cells have been shaded. Cells with a lower value than all their neighbours are blue; cells with a higher value than all their neighbours are red. Happy puzzling!

This puzzle was kindly provided by Brainfreeze puzzles, a puzzle making company run by the mathematician Laura Taalman and the software developer Philip Riley, whose book Colour Sudoku is available now.

The solution